arXiv:cond-mat/0210247 v1 10 Oct 2002

QUBITS AS SPECTROMETERS OF QUANTUM NOISE
R.J. SCHOELKOPF, A.A. CLERK, S.M. GIRVIN, K.W. LEHNERT and M.H. DEVORET
Departments of Applied Physics and Physics
Yale University, PO Box 208284, New Haven, CT 06520-8284

1. Introduction
Electrical engineers and physicists are naturally very interested in the noiseof circuits, amplifiers and detectors. This noise has many origins, some ofwhich are completely unavoidable. For example, a dissipative element (aresistor) at finite temperature inevitably generates Johnson noise. Engineers long ago developed spectrum analyzers to measure the intensity ofthis noise. Roughly speaking, these spectrum analyzers consist of a resonant
circuit to select a particular frequency of interest, followed by an amplifierand square law detector (e.g. a diode rectifier) which measures the meansquare amplitude of the signal at that frequency.
With the advent of very high frequency electronics operating at low
temperatures, we have entered a new regime ~ω > kB T , where quantum
mechanics plays an important role and one has to begin to think about
quantum noise and quantum-limited amplifiers and detectors. This topic is
well-studied in the quantum optics community and is also commonplace inthe radio astronomy community. It has recently become of importance inconnection with quantum computation and the construction of mesoscopic
electrical circuits which act like artificial atoms with quantized energy
levels. It is also important for understanding the quantum measurement
process in mesoscopic systems.
In a classical picture, the intensity of Johnson noise from a resistor vanishes linearly with temperature because thermal fluctuations of the charge
carriers cease at zero temperature. One knows from quantum mechanics,
however, that there are quantum fluctuations even at zero temperature,
due to zero-point motion. Zero-point motion is a notion from quantum mechanics that is frequently misunderstood. One might wonder, for example,
whether it is physically possible to use a spectrum analyzer to detect thezero-point motion. The answer is quite definitely yes, if we use a quantum

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system! Consider for example a hydrogen atom in the 2p excited state lying
3/4 of a Rydberg above the 1s ground state. We know that this state is
unstable and has a lifetime of only about 1 ns before it decays to the groundstate and emits an ultraviolet photon. This spontaneous decay is a naturalconsequence of the zero-point motion of the electromagnetic fields in thevacuum surrounding the atom. In fact, the rate of spontaneous decay gives
a simple way in which to measure this zero point motion of the vacuum.
Placing the atom in a resonant cavity can modify the strength of the noiseat the transition frequency, and this effect can be measured via a change
in the decay rate.
At finite temperature, the vacuum will contain blackbody photons which
will increase the rate of decay due to stimulated emission and also cause
transitions in the reverse direction, 1s → 2p, by photon absorption. With
these ideas in mind, it is now possible to see how to build a quantum
spectrum analyzer.
The remainder of this article is organized as follows. First we describe
the general concept of a two-level system as a quantum spectrum analyzer.
We next review the Caldeira-Leggett formalism for the modelling of adissipative circuit element, such as a resistor, and its associated quantum
noise. Then, a brief discussion of the single Cooper-pair box, a circuit which
behaves as a two-level system or qubit, is given. We then discuss the effects
of a dissipative electromagnetic environment on the box, and treat the caseof a simple linear, but nonequilibrium environment, consisting of a classicaltunnel junction which produces shot noise under bias. Finally, we describe
a theoretical technique for calculating the properties of a Cooper-pair box
coupled to a measurement system, which will be a nonlinear, nonequilibrium
device, such as a single-electron transistor. Equivalently, this allows one
to calculate the full quantum noise spectrum of the measurement device.
Results of this calculation for the case of a normal SET are presented.
2. Two-level systems as spectrum analyzers
Consider a quantum system (atom or electrical circuit) which has its twolowest energy levels 0 and 1 separated by energy E01 = ~ω01 . We suppose
for simplicity that all the other levels are far away in energy and can beignored. The states of any two-level system can be mapped onto the statesof a fictitious spin-1/2 particle since such a spin also has only two statesin its Hilbert space. With spin up representing the ground state (|gi) andspin down representing the excited state (|ei), the Hamiltonian is (taking
the zero of energy to be the center of gravity of the two levels)
H0 = −

~ω01
σz .
2

(1)

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In keeping with the discussion above, our goal is to see how the rate of ‘spinflip’ transitions induced by an external noise source can be used to analyze
the spectrum of that noise. Suppose for example that there is a noise sourcewith amplitude f (t) which can cause transitions via the perturbation1
V = Af (t)σx ,(2)

where A is a coupling constant. The variable f (t) represents the noise
source. We can temporarily pretend that f is a classical variable, although
its quantum operator properties will be forced upon us very soon. For now,
only our two-level spectrum analyzer will be treated quantum mechanically.
We assume that the coupling A is under our control and can be madesmall enough that the noise can be treated in lowest order perturbation
theory. We take the state of the two-level system to beαg (t)|ψ(t)i =.
(3)αe (t)In the interaction representation, first-order time-dependent perturbation
theory gives
Z
i t
(4)
dτ V̂ (τ )|ψ(0)i.
|ψI (t)i = |ψ(0)i −~ 0
If we initially prepare the two-level system in its ground state then theamplitude to find it in the excited state at time t isZiA tαe = −dτ he|σ̂x (τ )|gif (τ ) + O(A2 ),
(5)
~ 0
Z
iA t
dτ eiω01 τ f (τ ) + O(A2 ).
(6)
= −
~ 0
We can now compute the probability
Z Z
A2 t t2pe (t) ≡ |αe | = 2dτ1 dτ2 e−iω01 (τ1 −τ2 ) f (τ1 )f (τ2 ) + O(A3 )
~ 0 0

(7)

We are actually only interested on the average time evolution of the system
Z Z
A2 t tp̄e (t) = 2dτ1 dτ2 e−iω01 (τ1 −τ2 ) hf (τ1 )f (τ2 )i + O(A3 )
(8)
~ 0 0
1

The most general perturbation would also couple to σy but we assume that (as isoften, though not always, the case) a spin coordinate system can be chosen so that the
perturbation only couples to σx . Noise coupled to σz commutes with the Hamiltonianbut is nevertheless important in dephasing coherent superpositions of the two states. We
will not discuss such processes here.

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4
We can now perform a change of variables in the integrals, τ = τ1 − τ2 andT = (τ1 + τ2 ) /2, and we getp̄e (t) =

A2
~2

Z t
0

dT

Z B(T )

−B(T )dτ e−iω01 τ hf (T + τ /2)f (T − τ /2)i + O(A3 ) (9)whereB (T ) = T if T < t/2= t − T if T > t/2.
Let us now suppose that the noise correlation function is stationary (timetranslation invariant) and has a finite but small autocorrelation time τf .
Then for t  τf we can set the bound B (T ) to infinity in the last integral
and write
Z ∞
Z
A2 t
dτ e−iω01 τ hf (τ )f (0)i + O(A3 )(10)dTp̄e (t) = 2~ 0−∞
The integral over τ is effectively a sum of a very large number N ∼ t/τf ofrandom terms 2 and hence the value undergoes a random walk as a functionof time. Introducing the noise spectral density
Z +∞
Sf (ω) =dτ eiωτ hf (τ )f (0)i,(11)−∞

we find that the probability to be in the excited state increases linearlywith time,3A2(12)p̄e (t) = t 2 Sf (−ω01 )~The time derivative of the probability gives the transition rateΓ↑ =A2
Sf (−ω01 )
~2

(13)

Note that we are taking in this last expression the spectral density on thenegative frequency side. If f were a strictly classical source hf (τ )f (0)i would
2

The size of these random terms depends on the variance of f and on the value ofω01 τf For ω01 τf  1 the size will be strongly reduced by the rapid phase oscillations of
the exponential in the integrand.
3
Note that for very long times, where there is a significant depletion of the probabilityof being in the initial state, first-order perturbation theory becomes invalid. However, forsufficiently small A, there is a wide range of times τf  t  1/Γ for which Eq. 12 is valid.
Eqs. 13 and 14 then yield well-defined rates which can be used in a master equation todescribe the full dynamics including long times.

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be real and Sf (−ω01 ) = Sf (+ω01 ). However, because as we discuss below
f is actually an operator acting on the environmental degrees of freedom,[f (τ ), f (0)]6 = 0 and Sf (−ω01 )6 = Sf (+ω01 ).
Another possible experiment is to prepare the two-level system in its
excited state and look at the rate of decay into the ground state. The
algebra is identical to that above except that the sign of the frequency isreversed:A2(14)Γ↓ = 2 Sf (+ω01 ).
~We now see that our two-level system does indeed act as a quantum spectrum analyzer for the noise. Operationally, we prepare the system either
in its ground state or in its excited state, weakly couple it to the noise
source, and after an appropriate interval of time (satisfying the above inequalities) simply measure whether the system is now in its excited state or
ground state. Repeating this protocol over and over again, we can find theprobability of making a transition, and thereby infer the rate and hence thenoise spectral density at positive and negative frequencies. Note that in contrast with a classical spectrum analyzer, we can separate the noise spectral
density at positive and negative frequencies from each other since we canseparately measure the downward and upward transition rates. Negative
frequency noise transfers energy from the noise source to the spectrometer.
That is, it represents energy emitted by the noise source. Positive frequency
noise transfers energy from the spectrometer to the noise source.4 In order toexhibit frequency resolution, ∆ω, adequate to distinguish these two cases,
it is crucial that the two-level quantum spectrometer have sufficient phasecoherence so that the linewidth of the transitions satisfies the condition
ω01 /∆ω ≥ max[kB T /~ω01 , 1].
In thermodynamic equilibrium, the transition rates must obey detailedbalance Γ↓ /Γ↑ = eβ~ω01 in order to give the correct equilibrium occupanciesof the two states of the spectrometer. This implies that the spectral densities
obey
(15)
Sf (+ω01 ) = eβ~ω01 Sf (−ω01 ).
Without the crucial distinction between positive and negative frequencies,
and the resulting difference in rates, one always finds that our two level
4

Unfortunately, there are several conventions in existence for describing the noise
spectral density. It is common in engineering contexts to use the phrase ‘spectral density’
to mean Sf (+ω) + Sf (−ω). This is convenient in classical problems where the twoare equal. In quantum contexts, one sometimes sees the asymmetric part of the noise
Sf (+ω) − Sf (−ω) referred to as the ‘quantum noise.’ We feel it is simpler and clearer tosimply discuss the spectral density for positive and negative frequencies separately, since
they have simple physical interpretations and directly relate to measurable quantities.
This convention is especially useful in non-equilibrium situations where there is no simple
relation between the spectral densities at positive and negative frequencies.

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6systems are completely unpolarized. If, however, the noise source is anamplifier or detector biased to be out of equilibrium, no general relation
holds.
We now rigorously treat the quantity f (τ ) as quantum operator in theHilbert space of the noise source. The previous derivation is unchanged, andEqs. (13,14) are still valid provided that we interpret the angular brackets
in Eq. (8) as representing the quantum statistical expectation value for theoperator correlation (in the absence of the coupling to the spectrometer)Z +∞X
Sf (ω) =ραα hα|f (τ )|γihγ|f (0)|αi(16)dτ eiωτ−∞α,γwhere for simplicity we have assumed that (in the absence of the coupling
to the spectrometer) the density matrix is diagonal in the energy eigenbasis and time-independent (but not necessarily given by the equilibrium
expression). This yields the standard quantum mechanical expression forthe spectral density
Z +∞
X
i
(17)
Sf (ω) =dτ eiωτραα e ~ (α −γ )t |hα|f |γi|2
−∞

= 2π~

α,γ

X
α,γ

ραα |hα|f |γi|2 δ(γ − α − ~ω).

(18)

Substitution of this into Eqs. (13,14) we derive the familiar Fermi Golden
Rule expressions for the two transition rates.
In standard courses, one is not normally taught that the transition
rate of a discrete state into a continuum as described by Fermi’s Golden
Rule can (and indeed should!) be viewed as resulting from the continuum
acting as a quantum noise source. The above derivation hopefully provides
a motivation for this interpretation.
One standard model for the continuum is an infinite collection of harmonic oscillators. The electromagnetic continuum in the hydrogen atomcase mentioned above is a prototypical example. The vacuum electric field
noise coupling to the hydrogen atom has an extremely short autocorrelation time because the range of mode frequencies ωα (over which the~αdipole matrix element coupling the atom to the mode electric field E
is significant) is extremely large, ranging from many times smaller than thetransition frequency to many times larger. Thus the autocorrelation time ofthe vacuum electric field noise is considerably less than 10−15 s, whereas the
decay time of the hydrogen 2p state is about 10−9 s. Hence the inequalities
needed for the validity of our expressions are very easily satisfied.
Of course in the final expression for the transition rate, energy conservation means that only the spectral density at the transition frequency

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enters. However, in order for the expression to be valid (and in order forthe transition rate to be time independent), it is essential that there be awide range of available photon frequencies so that the vacuum noise has an
autocorrelation time much shorter than the inverse of the transition rate.
3. Quantum Noise from a Resistor
Instead of an atom in free space, we might consider a quantum bit capacitively coupled to a transmission line. The transmission line is characterized
by an inductance per unit length ` and capacitance per unit length c. A
semi-infinite transmission
line presents a frequency-independent impedancep
Z = R0 =`/c at its end and hence acts like an ideal resistor. The
dissipation is caused by the fact that currents injected at one end launchwaves which travel off to infinity and do not return. Very conveniently,
however, the system is simply a large collection of harmonic oscillators (thenormal modes) and hence can be readily quantized. This representation ofa physical resistor is essentially the one used by Caldeira and Leggett [1]in their seminal studies of the effects of dissipation on tunneling. The only
difference between this model and the vacuum fluctuations in free spacediscussed above is that the relativistic bosons travel in one dimension anddo not carry a polarization label. This changes the density of states as afunction of frequency, but has no other essential effect.
The Lagrangian for the system is
Z ∞
1
`
(19)
dx j 2 − q 2 ,
L=
2
2c
0
where j(x, t) is the local current density and q(x, t) is the local charge density. Charge conservation connects these two quantities via the constraint∂x j(x, t) + ∂t q(x, t) = 0.
We can solve this constraint by defining a new variableZ xθ(x, t) ≡dx0 q(x0 , t)

(20)

(21)

0

in terms of which the current density is j(x, t) = −∂t θ(x, t) and the charge
density is q(x, t) = ∂x θ(x, t). For any well-behaved function θ(x, t), thecontinuity equation is automatically satisfied so there are no dynamical
constraints on the θ field. In terms of this field the Lagrangian becomes
Z ∞
`
1
L=
(∂x θ)2(22)
dx (∂t θ)2 −2
2c0

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8
The Euler-Lagrange equation for this Lagrangian is simply√ the wave equation v 2 ∂x2 θ − ∂t2 θ = 0 where the mode velocity is v = 1/ `c.
¿From Eq. (21) we can deduce that the proper boundary conditions(in the absence of any coupling to the qubit) for the θ field are θ(0, t) =θ(L, t) = 0. (We have temporarily made the transmission line have a finite length L.) The normal mode expansion that satisfies these boundary
conditions is
r ∞
2X
kn πx
θ(x, t) =,(23)
ϕn (t) sin
LL
n=1

where ϕn is the normal coordinate and kn ≡ πn
L . Substitution of this form
into the Lagrangian and carrying out the spatial integration yields a set ofindependent harmonic oscillators representing the normal modes.
L=

∞
X
`

n=1

2

(ϕ̇n )2 −

1 2 2
k ϕ .
2c n n(24)

From this we can find the momentum pn canonically conjugate to ϕn andquantize the system to obtain an expression for the voltage at the end of thetransmission line in terms of the mode creation and destruction operators
r
r
∞
1X
21
~
kn
V =
∂x θ(0, t) =(a† + an ).
(25)
Lc
c n=1
2`Ωn nThe spectral density of voltage fluctuations is then found to be∞

SV (ω) = 2π

2 X ~Ωn{nγ (~Ωn )δ(ω + Ωn )+ [nγ (~Ωn )+ 1]δ(ω − Ωn )}, (26)
L
2c
n=1

where nγ (~ω) is the Bose occupancy factor for a photon with energy ~ω.
Taking the limit L → ∞ and converting the summation to an integral yields
SV (ω) = 2R0 ~|ω|{nγ (~|ω|)Θ(−ω) + [nγ (~ω) + 1]Θ(ω)},(27)

where Θ is the step function. We see immediately that at zero temperature
there is no noise at negative frequencies because energy can not be extracted
from zero-point motion. However there remains noise at positive frequencies
indicating that the vacuum is capable of absorbing energy from the qubit.
A more compact expression for this ‘two-sided’ spectral density of a
resistor is
2R0 ~ω
SV (ω) =,(28)
1 − e−~ω/kB T
which reduces to the more familiar expressions in various limits. For example, in the classical limit kB T  ~ω the spectral density is equal to therjsdraft12.tex; 10/10/2002; 18:02; p.8

9Johnson noise result5
SV (ω) = 2R0 kB T,(29)which is frequency independent, and in the quantum limit it reduces to
SV (ω) = 2R0 ~ωΘ(ω).

(30)

Again, the step function tells us that the resistor can only absorb energy,
not emit it, at zero temperature.
If we use the engineering convention and add the noise at positive andnegative frequencies we obtain
SV (ω) + SV (−ω) = 2R0 ~ω coth

~ω
2kB T

(31)

for the symmetric part of the noise, which appears in the quantum fluctuationdissipation theorem[2]. The antisymmetric part of the noise is simply
SV (ω) − SV (−ω) = 2R0 ~ω.

(32)This quantum treatment can also be applied to any arbitrary dissipative
network[3]. If we have a more complex circuit containing capacitors andinductors, then in all of the above expressions, R0 should be replaced by
ReZ(ω) where Z(ω) is the complex impedance presented to the qubit.
4. The Single Cooper-Pair Box: a Two-Level Quantum Circuit
The Cooper-pair box (CPB) is a simple circuit [4], consisting of a small
superconducting “island”, connected to a large reservoir via a single smallcapacitance Josephson junction, depicted as a box with a cross (Fig. 1). The
island is charge biased by applying a voltage (Vg ) to a nearby lead, called
the gate, which has a small capacitance to the island, Cg . The junction
is characterized by its capacitance, Cj , and its tunnel resistance, Rj . At
temperatures well below the transition temperature of the superconductor(TC ∼ 1.5 K for the usual Al/AlOx/Al junctions), none of the many (∼ 109 )quasiparticle states on the island should be thermally occupied, and thenumber of Cooper-pairs on the island is the only relevant degree of freedom.
We may then write the Hamiltonian for the box in terms of the states ofdifferent numbers of pairs on the island, which are eigenstates of the number
operator, n̂|ni = n|ni. The box Hamiltonian consists of an electrostatic
term, plus a Josephson term describing the coupling of the island to the5Note again that in the engineering convention this would be SV (ω) = 4R0 kB T .

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10
z
a)

z'

b)
|g><g|
Rj,Cj

Cg

Eel

θ

y
x

Vg

|e><e|

Z(ω)

EJ
x'

Figure 1. a) Circuit diagram of Cooper-pair box. b) Pseudo-spin representation of theenergies of Cooper-pair box. The density matrix for the two pure eigenstates lie along
the total effective field, collinear with the z’ axis.

lead,
H = Helectrostatic + HJosephson
X
EJ X(|n + 1ihn| + h.c.)= 4EC(n − ng )2 |nihn| −
2
n
n

(33)
(34)

The energy scale for the electrostatic interaction is given by the charging
energy, EC = e2 /2CΣ , where CΣ = Cj + Cg is the total island capacitance,
while the Josephson energy, EJ , is set by the tunnel resistance and the gapof the superconductor,
EJ =

∆ RK
h∆
=
.
8e2 Rj
8 Rj(35)

The electrostatic term is easily modulated by changing the voltage on thegate; the quantity ng = Cg Vg /2e that appears in the Hamiltonian corresponds to the total polarization charge (in units of Cooper pairs) injected
into the island by the voltage source.
This Hamiltonian leads to particularly simple behavior in the charge
regime, when the electrostatic energy dominates over the Josephson coupling, 4EC  EJ . In this case we can restrict the discussion to only two
charge states, |n = 0i and |n = 1i. For convenience we can reference theenergies of the two states to their midpoint, Emid = 4EC (1 − 2ng )2 , so thatthe Hamiltonian now becomes
1 −Eel −EJ(36)
H=
2 −EJ Eel
where Eel is the electrostatic energy that is now linear in the gate charge,
Eel = 4EC (1−2ng ). It is also now apparent that the Hamiltonian is identicalrjsdraft12.tex; 10/10/2002; 18:02; p.10

11
2

1.0

0

EJ

<n>

E/EC

|e>
4EC

0.5

|g>
-2
0.0

0.5
ngate

0.0
0.0

1.0

0.5
ng

1.0

Figure 2.
Energies (left) of ground and excited states of a Cooper-pair box with
EC = EJ vs. dimensionless gate charge, ng = Cg Vg /2e. The expectation value of acharge measurement, hni, (right) for the ground (solid line) and excited (dotted line)
states vs. ng .

to that of a fictitious spin-1/2 particle,
H=−

EJ
Eel
σz −
σx ,
2
2

(37)

under the influence of two psuedo-magnetic fields, Bz = Eel and Bx = EJ ,as depicted in Fig. 1. In other words, the box is a qubit or two-level system6 .
The state of the system is in general a linear combination of the states|n = 0i and |n = 1i. The state can be depicted using the density matrix,
which corresponds to a point on the Bloch sphere, where the north pole(+z-direction) corresponds to |n = 0i. The ground and excited states ofthe system will be aligned and anti-aligned with the total fictitious field,
i.e. in the ±z 0 directions.
It is also apparent from this discussion that the states of the box canbe easily manipulated by changing the gate voltage. The energies of theground and excited states, as a function of ng , are displayed in Figure 2.
The energy difference between the ground and excited bands varies from
4EC at ng = 0, 1, to a minimum at the charge degeneracy point, ng = 1/2.
At this point, the Josephson coupling leads to an avoided crossing, and thesplitting is EJ .
Also plotted is the expectation value of the number operator, h|n̂|i,which is proportional to the total charge on the island. In the geometrical6

Of course, this is an approximation, as there are other charge states (|n = 2i, etc.)
which are possible, but require much higher energy. Even outside the charge regime (i.e.
when EJ ≥ 4EC ) the two lowest levels of the box can be used to realize a qubit [5]. In
this case, the two states do not exactly correspond with eigenstates of charge, and matrix
elements are more complicated to calculate. Nonetheless, this regime can also be used as
an electrical quantum spectrum analyzer.

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12
picture of Fig. 1b, a measurement of charge (n̂) is equivalent to projecting
the state on the z-axis, n̂ = 21 (1 − σz ). We see that as the gate charge
is changed from 0 to 1, the ground state is initially |n = 0i, and thecharacter of the ground and excited states interchange on passing throughthe degeneracy point, leading to the transition between hn̂i = 0 and 1, whichis broadened by quantum fluctuations (the σx coupling). At the degeneracy
point, the ground and excited states lie in the ±x-directions, i.e. they are
symmetric and antisymmetric combinations of the two charge states. In
general, we will denote the ground and excited state of the CPB at aparticular gate voltage as |gi and |ei, which are given in terms of the charge
states by |gi = cos(θ/2)|0i+sin(θ/2)|1i and |ei = − sin(θ/2)|0i+cos(θ/2)|1i
respectively, where θ = arctan[EJ /Eel ] is a function of the gate voltage.
A nice property of the CPB in this regime is that the various matrix
elements can be calculated in a straightforward way. For example, theexpectation value of n̂ in the ground state, hg|n̂|gi, is therefore equal to
1/2(1 − hg|σz |gi) = sin2 (θ/2), from which we can find the ground statecharge as shown in Fig. 2. A perturbation in the gate charge, due forexample to a fluctuation or change in the applied gate voltage, will leadto a proportional change in the electrostatic energy, or the z-component of
the fictitious magnetic field. Such a perturbation will cause both dephasing
and transitions between states.
5. General Discussion of CPB Coupled to a Dissipative Environment
In the previous section we described the Hamiltonian and the eigenstates fora Cooper-pair box which is “charge-biased,” i.e. controlled with a voltage
applied to a gate capacitor Cg , as shown in Fig. 1. In our earlier treatmentof the box, the voltage and the dimensionless gate charge, ng were treatedas fixed parameters of the Hamiltonian (c-numbers). In this case, the box’s
evolution is purely deterministic and conservative. However, it is impossible,
even in principle, to control such a voltage with arbitrary precision at all
frequencies. In Fig. 1, the idealized source of the gate voltage is drawn inseries with an impedance Z(ω) of the gate lead. Generally, this gate lead
will be connected to external wiring (a transmission line), with a typicalreal impedance comparable to the impedance of free space (∼ 50 Ω) at
the microwave transition frequencies of the box. ¿From the fluctuationdissipation theorem we know that this impedance will introduce noise on
the gate voltage, even at zero temperature.
There are several effects of the voltage noise on the box, or the couplingof our spin-1/2 circuit to the many external degrees of freedom represented
by the gate impedance. First, even at zero temperature, we will find a finite
excited state lifetime, T 1, for the box. Second, at finite temperature, we willrjsdraft12.tex; 10/10/2002; 18:02; p.12

13
find a finite polarization of our psuedo-spin, i.e. some steady-state probability to find the spin in its excited state. Finally, the gate noise introduces
a random effective field felt by the spin, and a loss of phase coherence for a
superpostition state. It is this last effect which is most important in making
high-fidelity qubits and performing quantum computations, but it is thefirst two which depend most explicitly on the quantum nature of the noise.
We deal in this manuscript with only these first two features of the box’s
coupling to the electromagnetic environment, and ignore the dephasing7 . Of
course, the other parameter in the Hamiltonian, the Josephson energy, canin principle fluctuate, especially as in many experiments the box’s junctionis split into a small SQUID in order to provide external tuning of EJ with
an applied flux. We concentrate here only on the voltage noise (fluctuationsin the σz part of the Hamiltonian) for simplicity.
We begin with a very simple treatment of the dynamics of the two-levelsystem under the influence of the gate voltage noise. We are interested in
the ensemble-averaged behavior of our psuedo-spin, which is best describedusing the density matrix approach, and is detailed in Section 7 on the
coupling of the box to a measuring SET. The basic effects, apart fromdephasing, however, can be captured simply by examining the probabilities
pg and pe of finding the box in its ground (|gi) or excited (|ei) states.
The noise of the external environment can drive transitions from ground toexcited state and vice-versa, at rates Γ↑ and Γ↓ , respectively. The coupledmaster equations for these probabilities aredpe= p g Γ↑ − p e Γ↓dtdpg
= p e Γ↓ − p g Γ↑
dt

(38)
(39)

Of course conservation of probability tells us that pe + pg = 1, so we
introduce the polarization of the spin-1/2 system, P = pg − pe . In steadystate, the detailed balance condition is pe Γ↓ = pg Γ↑ . The two rates Γ↑and Γ↓ are related by Equations 13 and 14 to the spectral densities of thenoise at negative and positive frequencies. We see immediately that if thespectral density is symmetric (classical!), then the rates for transitions up
and down are equal, the occupancies of the two states are exactly equal, andthe polarization of the psuedo-spin is identically zero. It is the quantum, or
antisymmetric, part of the noise which gives the finite polarization of the
spin. Even in NMR, where the temperature is large compared to the level
splitting (~ω01 ≤ kB T ), this effect is well-known and crucial, as the small
but non-zero polarization is the subject of the field!
7

For a nice recent treatment of dephasing in Josephson junctions, see Ref. [6].

rjsdraft12.tex; 10/10/2002; 18:02; p.13

14
Solving for the steady-state polarization, we findPss =Γ↓ − Γ↑S(+ω01 ) − S(−ω01 )=Γ↓ + Γ↑S(+ω01 ) + S(−ω01 )(40)

An measurement of the steady-state polarization allows one to observe theamount of asymmetry in the noise, so the two-level system is a quantum
spectrum analyzer.
If we can create a non-equilibrium polarization, P = Pss + ∆P (a purestate is not necessary) of our two-level system, we expect it to return to thesteady state value. Substituting the modified probabilities pe (t) = pess −∆P (t)/2 and pg (t) = pgss + ∆P (t)/2 into our master equations above, wefind an equation for the deviation of the polarizationd(∆P (t))= −∆P (t)(Γ↑ + Γ↓ ).
dt

(41)

Thus the system decays to its steady-state polarization with the relaxation
rate Γ1 = Γ↑ + Γ↓ = (A/~)2 [S(−ω01 ) + S(+ω01 )] related to the totalnoise at both positive and negative frequencies. In NMR, the time 1/Γ1
is referred to as T 1. In the zero-temperature limit, there is no possibility ofthe qubit absorbing energy from the environment, so Γ↑ = 0, and we findfull polarization P = 1, and a decay of any excited state population at a
rate Γ↓ = 1/T 1 which is the spontaneous emission rate.
It is worth emphasizing that a quantum noise source is always characterized by two numbers (at any frequency), related to the positive andnegative frequency spectral densities, or to the symmetric and antisymmetric parts of the noise. These two quantities have different effects on a
two-level system, introducing a finite polarization and finite excited-state
lifetime. Consequently, a measurement of both the polarization and T 1 of a
two-level system is needed to fully characterize the quantum noise coupled
to the qubit. Such measurements in electrical systems are now possible,
and some of us [7] have recently performed such a characterization forthe specific case of a CPB coupled to a superconducting single-electrontransistor.
Our discussion in this section uses the language of NMR to describe
the effects on the two-level system. There are, however, several possible
protocols8 for measuring the quantum noise, and several different “basissets” or measured quantities which describe the noise process or the quantum reservoir to which the two-level system is coupled. Table 1 contains
a “translation” between the specific pairs of quantities that are commonly
8

The idea of watching the decay from the pure states |ei and |gi to measure
SV (±|ω01 |) separately was described in Section 2.

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15
TABLE I.

Different ways to characterize a quantum reservoir.
2

Fermi Golden RuleS (+|ω|)Γ↑ (ω) = A~2 V2Γ↓ (ω) = A
S (−|ω|)~2 V

Fluct.-Diss. Relationnγ (ω) = 2Γ↑ /(Γ↓ − Γ↑ )Re[Z(ω)] = A~2 ω (Γ↓ − Γ↑ )NMR

T 1 = (Γ↓ + Γ↑ )−1

P = (Γ↓ − Γ↑ )/(Γ↓ + Γ↑ )Quantum Optics

BEinstein = Γ↑

AEinstein = Γ↓ − Γ↑used in different disciplines, and their relation to the positive and negative
noise spectral densities. In all cases, though, two separate numbers are
required to specify the properties of a quantum reservoir.
6. The Box Coupled to an Ohmic Environment
We can now proceed to the effects of a specific dissipative coupling to the
Cooper-pair box. The noise on the gate voltage will lead to a fluctuation of
the gate charge parameter, ng , and thus to a fluctuation of the electrostatic
energy, i.e. the σz term in the Hamiltonian (Eq. 37). Depending on the
average value of ng , this fluctuation will consist of fluctuations which areboth longitudinal (k to σz0 ) and transverse (⊥ to σz0 ). To calculate the ratesof transitions between the states |ei and |gi, we need to find the couplingstrength A of this perturbation in the σx0 direction. Referring to Fig. 1,
we see that σz = cos(θ)σz0 − sin(θ)σx0 . If we let the gate charge now beng (t) = n¯g + δng (t), we may rewrite the Hamiltonian Eq. 37 in the new
eigenbasis as
H=−

E01 0
σ + 4EC cos(θ)δng (t)σz0 − 4EC sin(θ)δng (t)σx0 .
2~ z

(42)

The time-varying term in the σz0 direction will effectively modulate thetransition frequency, ω01 = E01 /~, and cause dephasing. In terms of thegate voltage noise, V (t), the σx0 perturbation term has the form AV (t)σx0 =eκ sin(θ)V (t)σx0 , where e is the electron’s charge and κ = Cg /CΣ is thecapacitive coupling. Using Eq. 14, we find
 e 2Γ↓ =κ2 sin2 (θ)SV (+ω01 ).
(43)~
If the environment is effectively at zero temperature (~ω01  kB T ), then
SV (+ω01 ) = 2~ω01 R, and the quality factor of the transition is
Q = ω01 /Γ↓ =RK
1
2 4πR ,
2
κ sin θ(44)

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16
where RK = h/e2 is the resistance quantum.
For a finite temperature, we have rates in both directions, and thepolarization of the psuedo-spin is given by the ratio of the antisymmetric(Eq. 32) to symmetric (Eq. 31) spectral densities, P = tanh(~ω01 /2kB T ),as one expects for any two-level system at temperature T . An example of
the average charge state of a Cooper-pair box at finite temperature, and of
the polarization and equilibration time T1, are shown in Figure 3. As thegate voltage is varied, the transition frequency of the box changes from amaximum near ng = 0, to a minimum ω01 = EJ /~ at the degeneracy pointng = 0.5 and then back again. We see that the states of the box are generally
most “fragile” near the avoided crossing. First, the energy splitting is a minimum here, leading to the lowest polarization of the psuedo-spin. Second,
because the eigenstates |gi and |ei point in the σx direction, the matrix
elements for the voltage fluctuations of the environment are maximal, i.e.
the noise is orthogonal to the spin. This also implies that the dephasing
effects are minimal at this degeneracy point, which offers great advantages
for improving the decoherence times [5], but the excited state lifetimes are
smallest at this point. One also sees that the lifetimes become large awayfrom the degeneracy, where the electrostatic energy dominates over theJosephson energy, which offers advantages when measuring the charge state.
In the limit that EJ could be suppressed to zero during a measurement, thematrix elements (for voltage noise) vanish, and a quantum non-demolition(QND) measurement [8] could be performed. The idea of using the qubit
as a quantum spectrum analyzer is precisely the reverse, where we measure
the “destruction” in the two-level system (i.e. inelastic transitions causedby the coupling of the states to the environment), in order to learn about
the quantum noise spectrum of the fluctuations.
The Cooper-pair box can of course also be used to measure the more interesting spectral densities of nonequilibrium devices. The simplest example
is to replace the gate resistance by a tunnel junction. If we arrange to bias
the junction using, e.g. an inductor, a dc current I and an average dc voltage
V can be maintained across the junction. Classically, the current noise ofsuch a tunneling process is frequency independent, SI = 2eI. The voltage
noise density presented to the CPB’s gate would then be SV = 2eIRT2 ,where RT is the junction tunnel resistance. In fact, this “white” spectral density can only extend up to frequencies of order ω = eV /~, themaximum energy of electrons tunneling through the junction. The correct
high-frequency form of the symmetrized noise density was calculated by
Rogovin and Scalapino [9],~ω − eV~ω + eV+ R (~ω − eV ) coth,SV (ω) = R (~ω + eV ) coth
2kB T
2kB T
(45)

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17

(ω<0)

Emission(ω>0)

SV (ω)

1.0<n>a) Absorption

0.5
0.0
0.0

ω=0

1.0

b)

c)

1.0

40

<P> 0.5

0.0
0.0

0.5
ng

20

0.5
ng

T1 (µs)0
1.0

Figure 3. The box coupled to an equilibrium, Ohmic environment, i.e. a resistor. a)
Two-sided noise spectral density, of the voltage, SV (ω), for a resistor at T=0 (solidline) and finite temperature (dashed line) b) Average charge of box with EC = 1K,
EJ = 0.5K when coupled with strength κ = 0.01 to a resistor with resistance R = 50 Ωand temperature T = 0.5K. c) Polarization (dotted line) and relaxation time T1 for thesame parameters. Full line is the rate of spontaneous emission, i.e. T1 at zero temperature.

and was indirectly measured in a mesoscopic conductor using a conventional
spectrum analyzer [10]. This noise can also be expressed [11] in its two-sidedform(~ω − eV ) RT(~ω + eV ) RT+,
(46)
SV (ω) =− ~ω+eV− ~ω−eV
1 − e kB T
1 − e kB T

and is displayed in Fig. 4. Notice that the antisymmetric part of this noise is
the same as that of the ordinary resistor, SV (+ω) − SV (−ω) = 2~ωRT , andis independent of the voltage. Also shown in Fig. 5 is the polarization andrelaxation time, T1, of a CPB coupled to a shot noise environment. We seethat full polarization is achieved only when ~ω01  eV . For low transition
frequencies, the polarization is inversely proportional to the current throughthe junction. Aguado and Kouwenhoven [11] have described the use of adouble quantum dot as a two-level system to probe this behavior of theshot noise.
Given our discussion so far, it is now interesting to ask what the effects
of a real quantum measurement on a quantum circuit will be. A quantum
measurement device will in general be neither linear, Ohmic, nor equilib-

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18

Emission(ω>0)

SV (ω)

1.0<n>Absorption
(ω<0)

0.5
0.0
0.0

ω=0

0.5
ng

1.0

1.0

40
<P> 0.5

0.0
0.0

20

0.5
ng

T1 (µs)0
1.0

Figure 4.
The box coupled to an nonequilibrium, Ohmic environment, i.e. a 50 Ω
tunnel junction. a) Two-sided noise spectral density, of the voltage, SV (ω), for a junction
at T=0.02 K, with zero voltage (dotted line), and increasing bias voltages (solid anddashed lines). b) Average charge of box with EC = 1K, EJ = 0.5K when coupled withstrength κ = 0.01 to a junction biased at eV = 1.5 K. c) Polarization (dotted line) andrelaxation time T1 for the same parameters (solid) and for T=0, V=0 (dashed line).

rium. Obviously, if we hope to characterize this measurement process, andto understand what one will observed when the qubit is coupled to thenoise processes of the measuring device, we will need to calculate the fullquantum (two-sided!) noise spectrum of the amplifier or detector.
7. Single-Electron Transistor Coupled to a Two-Level System
We have seen in previous sections that a two-level system (TLS) may be
used as a “spectrum analyzer” to measure quantum noise. Here, we use thistechnique to theoretically calculate quantum noise. Instead of simply studying the “noisy” system of interest in isolation, one can study a composite
system consisting of the “noisy” system coupled to a TLS; by calculating
the relaxation and excitation rates of the TLS, one can efficiently calculate the quantum noise of interest9 . We demonstrate the usefulness of this9

Note that the spirit of our approach is similar to that employed in the theory of fullcounting statistics [12]. There too one attaches an auxiliary spin 1/2 to the scatteringsystem of interest, and studies the dynamics of the fully coupled system to obtain therjsdraft12.tex; 10/10/2002; 18:02; p.18

19
technique by outlining a calculation of the quantum charge noise of a single
electron transistor (SET). This is an important example, as when an SET isused as an electrometer, it is this noise which determines the measurement
backaction.
The SET consists of a metallic island attached via tunnel junctions tosource and drain reservoirs. It is described by the Hamiltonian:X
HSET =(εk − µα ) c†kα ckα + EC (n − N )2 + HT
(47)k,α=L,R,I

HT = tXk,q,α=L,RF † c†kI cqα + h.c.

(48)

The first term in HSET describes the kinetic energy of electrons in the leads(α = L, R) and on the island (α = I). The second term is the Coulomb
charging energy which depends on n, the number of excess electrons on
the island. This interaction term can be tuned by changing the voltage
on a nearby gate electrode which is capacitively coupled to the island; N
represents the dimensionless value of this voltage. Finally, HT describes thetunneling of electrons through the two SET tunnel junctions; the conductance of each junction (in units of e2 /h) is given by g = 4π 2 t2 ν02 , with ν0†
being the density of† states. F is an auxiliary operator which increases n†
by one: n, F = F . For simplicity, we assume that the two junctions of
the SET are completely symmetric (i.e. equal junction capacitances anddimensionless conductances).
Throughout this section, we will be interested in the regime of sequential
tunneling in the SET, where transport involves energy-conserving transitions between two charge states of the SET island, say n = 0 and n = 1.
These transitions are described by simple rates, which can be derived via
Fermi’s Golden rule:Γαn±1,n = γ([∆E]αn±1,n )
g∆E/h
BT )
1 − e−∆E/(k


 
1
1
α
∆En±1,n = ∓2EC n ± − N ±− δα,R eVDS22γ(∆E) =

(49)
(50)
(51)

Γαn±1,n is the tunneling rate from the charge state n to n ± 1 throughjunction α; ∆E is the energy gained in making the tunneling transition,and includes contributions both from the drain-source voltage VDS andfrom the charging energy. Sequential tunneling is the dominant transportmechanism when the junction conductances are small (i.e. g/(2π)  1),statistics of charge transfer in the scatterer.

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20
and the dimensionless gate voltage N is not too far away from a charge
degeneracy point. Sequential tunneling is the most important regime formeasurement applications, as it yields the largest SET currents.
At low temperatures, only tunnel events which follow the voltage arepossible. There are thus there only two relevant rates: n = 0 → 1 transitions
occur through the left junction at a rate ΓL
10 , while n = 1 → 0 occur throughR
the right junction at a rate Γ01 . The average current will be given by:
hIi = eΓ̄ ≡ eRΓL
10 Γ01R
ΓL
10 + Γ01(52)

We are interested in calculating SQ (ω), the quantum noise associatedwith fluctuations of the charge on the central island of the SET. It is definedas:Z ∞SQ (ω) =dthn(t)n(0)ie−iωt(53)−∞Note that we can equivalently think of SQ as describing the voltage noiseof the island, as Visland = en/CΣ , where CΣ is the total capacitance of theisland. In two limits, the form of the island charge noise can be anticipated.
For ω → 0, the noise will correspond to classical telegraph noise– the island
charge n fluctuates between the values 0 and 1, with Poisson-distributed
Rwaiting times determined by the rates ΓL
10 and Γ01 . We thus expect asymmetric, Lorentzian form [13] for the noise at low frequencies:
SQ (ω) →2Γ̄
R 2
ω 2 + (ΓL
10 + Γ01 )(ω  EC )(54)

For large frequencies |ω|  EC , we expect that correlations due to thecharging energy will have no influence on the noise. The system will effectively look like two tunnel junctions in parallel, and we can use the resultsof Sec. 6 for the corresponding voltage noise. Noting that each junction
effectively consists of a resistor and capacitor in parallel, we have at zerotemperature:
CΣ2
CΣ2
×
S
(ω)
→
[2~ωRe Ztot Θ(ω)]
V
e2 
e2
g
ω
= 4
Θ(ω)

2
2π
g 4EC
2
+ω
2π ~

SQ (ω) =(|ω|  EC )(55)

Note that SQ (ω) decays as 1/ω at large frequencies, whereas Eq. (54) for
classical telegraph noise decays as 1/ω 2 .
Given these two limiting forms, the question now becomes one of how
the SET interpolates between them. One might expect that the two results

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21
should simply be added in quadrature, but as with combining thermal andquantum noise (see Sec. 6), this is approach is too simple. A completely
quantum mechanical way of calculating the noise for any frequency isneeded. This was recently accomplished by Johansson et. al [14] using an
extension of a technique developed by Schöller and Schön [15]. Here, we rederive their results using the coupled system approach outlined above. Thismethod is physically motivated and allows for a heuristic interpretation of
the final result.
8. SET Coupled to a Qubit
We now consider a system where the SET is coupled to a two-level system(i.e. a qubit), with a coupling Hamiltonian which can induce transitionsin the TLS. Using spin operators to describe the qubit, and assuming
operation at the degeneracy point for simplicity, where the transitions are
fastest10 , we have:1(56)
H = HSET − Ωσx + Aσz n,2where Ω is the qubit splitting frequency11 , and A is the coupling strength.
We can define the rates Γ↑ and Γ↓ which are, respectively, the rate at which
the qubit is excited by the SET, and the rate at which the qubit is relaxedby the SET. For a weak coupling (A → 0), one has (c.f. Eq. 14,13 in Sec. 2):
Γ↓/↑ =A2
SQ (±Ω)~(57)

Eq. (57) tells us that if we know the rates Γ↑ and Γ↓ for a weakly coupledsystem at an arbitrary splitting frequency Ω, we know the quantum noise
SQ (Ω) at all frequencies. This is the essence of the technique previously
described, in which a qubit acts as a quantum spectrum analyzer of noise.
Here, we mimic this approach theoretically by obtaining Γ↑ and Γ↓ from a
direct analysis of the coupled system in the limit of weak coupling (A → 0).
The object of interest is the reduced density matrix ρ which describes boththe charge n of the transistor island and the state of the qubit. We areinterested in two quantities. First, what is the stationary state of the qubit?
The stationary populations of the two qubit states (which are determined
10
This amounts to maximizing the “destruction” due to the SET’s noise, and thecase where θ = π/2, the qubit eigenstates are in the σz0 = σx direction, and the SET’s
perturbation is in the −σx0 = σz direction (c.f. Eq. 42). After the noise of the SET isfound, we can then recalculate the effects on the qubit at various ng or values of θ byincluding the modified matrix elements in the coupling coefficient, A.
11
Henceforth we use Ω for the transition frequency, instead of the previous notation
ω01 = E01 /~, for compactness.

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22
from the time-independent solution for ρ) will tell us the polarization of
the qubit, and the amount of asymmetry in the noise (c.f. Eq.40). Second,
how quickly do the qubit populations relax to their stationary value? Thisrelaxation will be described by a time-dependent solution of ρ characterized
by a mixing rate Γmix which is the sum of Γ↑ and Γ↓ (c.f. Eq. 41). Fromthese two results we can solve for the individual values of Γ↑/↓ .
To deal with the dynamics of ρ, we make use of the fact that sequential
tunneling processes are completely described by lowest-order perturbation
theory in the tunneling Hamiltonian HT . Keeping only second order terms(there are no non-vanishing first order terms), one obtains the following
standard evolution equation in the interaction picture:1dρ(t) = −
dt
~

Z t

−∞




dt0 h HT (t), HT (t0 ), ρ(t0 ) ⊗ ρF i

(58)

The angular brackets denote the trace over the single-particle degrees of
freedom in the SET leads and island; ρF is the equilibrium density matrix
corresponding to the state of these degrees of freedom in the absence oftunneling.12 Note that a similar density matrix analysis of a qubit coupled
to a SET was recently discussed by Makhlin et. al [16]; unlike the present
case, these authors restricted attention to a vanishingly small splitting
frequency Ω.
To make progress with Eq. (58), we make a Markov approximation,
which involves replacing ρ(t0 ) on the right-hand side with ρ(t). This ispermissible as we are interested in the slow dynamics of ρ. We want to findboth the stationary solution of ρ, for which the Markov approximation isexact, and the mixing mode, a mode whose time dependence is ∝ e−(Γ↑ +Γ↓ )t .
This mode is also arbitrarily slow in the weak coupling limit A → 0 ofinterest. Finding the stationary mode and the mixing mode correspond toevaluating the polarization and T1 of the qubit (c.f. Eq. 40 and Eq. 41), as
was shown earlier for the master equation of the probabilities in Section 5.
Note that the Markov approximation should be made in the Schrödinger
picture, as it is in the Schrödinger picture that ρ will be nearly stationary(i.e. all oscillations associated with the qubit splitting frequency Ω will be
damped out in the long-time limit).
Evaluation of Eq. (58) in the Markov approximation results in the appearance of rates which are generalizations of those given in Eq. (49). Now,
however, these rates depend on the initial and final state of the qubit–
tunneling transitions can simultaneously change both the charge state of
the SET island and the state of the qubit. The resulting equation is most
12

In the diagrammatic language of Ref. [15], Eq. (58) is equivalent to keeping all (HT )2
terms in the self-energy of the Keldysh propagator governing the evolution of ρ.

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23
easily presented if we write the reduced density matrix ρ in the basis of
eigenstates at zero tunneling. For each value of island charge n, there is adifferent qubit Hamiltonian, and correspondingly a different a qubit groundstate |gn i and excited state |en i. When a tunneling event occurs in the SET,
there is a sudden change in the qubit Hamiltonian. As the qubit groundand excited states at different values of n are not orthogonal, tunnelingtransitions in the SET are able to cause “shake-up” transitions in the qubit.
In the limit A → 0, the relevant matrix overlaps are given by:
1 A(m − n) 2
hgm |gn i = 1 −= hem |en i(59)2ΩA(m − n)(60)
hem |gn i =ΩDefining the frequency dependent rate Γn±1,n (ω) as:XΓn±1,n (ω) ≡γ([∆E]αn±1,n + ~ω),
(61)
α=L,R

where ∆E and γ(∆E) are defined in Eqs. (51) and (50), the required tunnel
rates take the form:
Γm,n ≡ Γm,n (0)Γ±m,n ≡ Γm,n (±Ω)(62)

The Γ+ rates correspond to tunneling events where the qubit is simultaneously relaxed, and thus there is an additional energy Ω available fortunneling. For large Ω, tunneling processes which are normally energetically forbidden can occur if they are accompanied by qubit relaxation.
Similarly, the Γ− rates describe tunnelling events where the qubit is simultaneously excited, with the consequence that there is less energy available
for tunneling.
Returning to the evolution equation Eq. (58), note that we do notneed to track elements of ρ which are off-diagonal in the island charge
index n– there is no coherence between different charge states, as tunneling
events necessarily create an electron-hole excitation. Further, if we focus
on small qubit frequencies, we may continue to restrict attention to onlyn = 0 and n = 1 (i.e. Ω is not large enough to “turn on” tunnelingprocesses which are normally energetically forbidden). Thus, there are 8
relevant matrix elements of ρ– for each of the four qubit density matrix
elements (i.e. gg, ee, ge, eg), there are two possible island charge states.
We combine
these elements into a vector ρ~ = (ρgg , ρee , ρge , ρeg ), whereρgg = h0, g0 |ρ|0, g0 i, h1, g1 |ρ|1, g1 i , etc. Organizing the resulting evolution
equation in powers of the coupling A, we obtain in the Schrödinger picture:
A
A2
d
ρ
ρ
~ = (Λ0 + Λ1 + 2 Λ2 + ...)~
dt
Ω
Ω

(63)

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24
We discuss the significance of the matrices Λj in what follows.
The 8 × 8 matrix Λ0 describes the evolution of the system at zero
coupling:


M 0
0
0

 0 M
0
0
,
(64)
Λ0 = 

 0 0 +iΩ + M 00
0 00−iΩ + M 0with the 2 × 2 matrices M and M 0 being defined by:
1 −(Γ+−Γ10 Γ01
+ Γ−
) Γ+
+ Γ−
0
10
10
01
01
M=
(65)
M =
−
−
Γ10 −Γ01Γ+−(Γ+
2
10 + Γ10
01 + Γ01 )At zero coupling there are no transitions between different qubit states, andhence Λ0 has a block-diagonal form. There are two independent stationary
solutions of Eq. (63) at A = 0 (i.e. two zero eigenvectors of Λ0 ), whichcorrespond to being either in the qubit ground or qubit excited state:~zg = (p0 , p1 , 0, 0, 0, 0, 0, 0) ,~ze = (0, 0, p0 , p1 , 0, 0, 0, 0) .

(66)

(p0 , p1 ) are the stationary probabilities of being in the n = 0 or n = 1
charge states:


Γ10
Γ01
,
(67)
(p0 , p1 ) =Γ01 + Γ10 Γ01 + Γ10
The existence of two zero-modes is directly related to the fact that at zero
coupling (A = 0), the probabilities to be in the qubit ground and excitedstate are individually conserved.
At non-zero coupling, the matrices Λ1 and Λ2 appearing in Eq. (63)generate transitions between different qubit states. The matrix Λ2 directly
couples ρgg and ρee , while Λ1 couples ρgg and ρee to the off-diagonal blocks
ρge and ρeg . The effect of these matrices will be to break the degeneracy of
the two zero modes of Eq. (63) existing at A = 0. After this degeneracy isbroken, there will still be one zero mode ρ0 , describing the stationary stateof the coupled system (the existence of a stationary solution is guaranteedby the conservation of probability). For weak coupling, the qubit density
matrix obtained from ρ0 will be diagonal in the basis {|ghni i, |ehni i}, whichcorresponds to the average SET charge hni = p1 . The ratio of the occupancies of these two qubit states will yield the ratio between the relaxation
rate Γ↓ and the excitation rate Γ↑ . In addition, there will also be a slow,
time-dependent mode of Eq. (63) arising from breaking the degeneracy of
the two A = 0 zero modes. This time-dependent mode will describe how
a linear combination of zg and ze relaxes to the true stationary state, andwill have an eigenvalue λ = −Γ↑ − Γ↓ , i.e. the mixing rate.

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25
Thus, we need to do degenerate second order perturbation theory inthe coupling A to obtain the relaxation and excitation rates Γ↓ and Γ↑ .
The only subtlety here is that the matrix M is not Hermitian, implyingthat it has distinct right and left eigenvectors. Letting ~z̃ represent the left
eigenvector of Λ0 corresponding to the right eigenvector ~z, we define theprojector matrix P as:
P = |zg ihz̃g | + |ze ihz̃e |,(68)

and let P⊥ denote 1 − P. As usual, degenerate second order perturbation theory requires diagonalizing the perturbation in the space of thedegenerate eigenvectors. We are thus led to look at the matrix Q, defined
as:

A2 
(69)
Q = 2 PΛ2 P + PΛ1 P⊥ [−Λ0 ]−1 P⊥ Λ1 PΩ
¿From the definition of Q, we may immediately identify the rates Γ↑ andΓ↓ :Γ↑ = hz̃e |Q|zg iΓ↓ = hz̃g |Q|ze i(70)

We thus see how the rates Γ↑,↓ arise in the present approach– they
are related to breaking the degeneracy between two zero-modes (stationary
solutions) which exist at zero coupling. Note that there are two distinct
contributions to Γ↑,↓ , coming from the two terms in the matrix Q: a “direct”
contribution involving Λ2 and an “interference” contribution involving Λ1
acting twice. These two terms have a different physical interpretation, as
will become clear.
Let us first consider the rate Γ↑ , which describes how noise in the SET
causes ground to excited state transitions in the qubit. For this rate, ourapproximation of only keeping two charge states will be valid for all splitting
frequencies Ω. To evaluate the “direct” contribution to this rate, which
involves the first term in the matrix Q, note that the relevant part of Λ2
has the expected form:


0 Γ−
01
Λ2 |ee,gg =
(71)
Γ−
10 0
i.e. it consists of tunnel rates which correspond to having given up an energyΩ to the qubit. Using Eqs. (69) and (70), we find: 2
 2 − AA
Γ01 Γ−−−
10 + Γ10 Γ01Γ↑ |direct =p0 Γ10 + p1 Γ01 =(72)ΩΩ
Γ10 + Γ01
The direct contribution to Γ↑ has a very simple form: for each charge staten = 0, 1, add the rate to tunnel out of n while exciting the qubit, weighted

rjsdraft12.tex; 10/10/2002; 18:02; p.25

26
by both the probability to be in state n, and the overlap between groundand excited states (i.e. (A/Ω)2 ). This is very similar to how one typicallycalculates the current for a SET: one adds up the current associated witheach charge state (i.e. a difference of rates), weighted by the occupancyof the state. The direct contribution to Γ↑ neglects any possible coherence
between successive excitation events; as a result, it fails to recover theclassical expression of Eq. (54) in the small-Ω limit.
We now consider the “interference” contribution to Γ↑ coming fromthe second term in the expression for matrix Q (c.f. Eq. (69)). After somealgebra, we obtain the following for the interference contribution to Γ↑ :
Γ↑ |int = − (ΓΣ )2
2A2
−
−
p
Γ
+
p
Γ
0 10
1 01
Ω2
Ω2 + (ΓΣ )2(73)

where:+−+
Γ−
10 + Γ10 + Γ01 + Γ01(74)2This contribution is purely negative, and is only significant (relative tothe direct contribution) at low frequencies Ω < Γ. We can interpret thisequation as describing the interference between two consecutive excitation
events. For example, consider the first term in Eq. (73). This describes
a process where a SET initially in the charge state n = 0 undergoes atunnel event to the n = 1 state, creating a superposition of qubit groundand excited states. At some later time the SET relaxes to the stationary distribution (p0 , p1 ) of the charge states, again partially exciting thequbit. Letting ∆t represent the time between these two events, we have theapproximate sequence:

ΓΣ ≡

Γ−

10
|0, g0 i −→|1, g1 i + α|1, e1 i∆t−→ eiΩ∆t/2 |1, g1 i + e−iΩ∆t/2 α|1, e1 i


ΓΣ
−→
eiΩ∆t/2 β − e−iΩ∆t/2 α |0, e0 i + ...

(75)
(76)

Here, α is the amplitude associated with qubit excitation having occurred
during the first (n = 0 → 1) tunnel event, while β is the amplitude
associated with excitation occurring during the second (n = 1 → 0) tunneling. These amplitudes will be given by the corresponding matrix overlap
elements:AA
β = he0 |g1 i ' −(77)
α = he1 |g0 i 'ΩΩIn the final state after the two tunnelings (Eq. (76)), there are two terms
in the amplitude of the state |0, e0 i, corresponding to the fact that qubit
excitation could have occurred in either the first or the second tunnel event.

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27
To get a rate for this double excitation event, we should take the modulus
squared of the final |0, e0 i state amplitude, then multiply by the occupancyof the initial state (p0 ) and the rate of the first tunnel event (Γ10 ). Theinterference term in the resulting expression takes the form:
2A2Γ↑ |int = (p0 Γ10 ) × 2Re α∗ βeiΩ∆t = −(p0 Γ10 ) 2 cos(Ω∆t)Ω(78)

The above expression is a function of the time ∆t between the first and second tunnel events. This time is determined by the fact that the intermediate
superposition state (Eq. (75)) corresponds to a non-stationary distributionof charge on the SET island, and will decay via tunneling to the stationary
distribution (p0 , p1 ) at a rate ΓΣ . Taking this decay to be Poissonian, andaveraging over ∆t, we obtain:Γ↑ |int = −(p0 Γ10 )

2A2 (ΓΣ )2Ω2 Ω2 + (ΓΣ )2(79)This is precisely the first term in Eq. (73); the second term can be obtained in the same way, by now considering a situation where the SET isinitially in the n = 1 charge state. As claimed, Γ↑ |int corresponds to theinterference between two consecutive excitation events. The negative signof this contribution can be directly traced to the matrix overlap elements(c.f. Eq. 77). Also, we see that the suppression of the interference term atlarge Ω results from phase randomization occurring during the delay time
between the two excitation events.
Returning to the total noise, we combine Eq. (73) with the direct contribution Eq. (72) to Γ↑ ; comparing against Eq. (57), we obtain the final
expression for SQ (Ω) at all negative frequencies:
SQ (−|Ω|) =−p 0 Γ−
10 + p1 Γ01+−+Ω2 + 14 Γ−
10 + Γ10 + Γ01 + Γ01

2

(80)

α , ∆E α ) ' V
Note for large |Ω| (i.e. |Ω| > max(∆E01
DS /2), SQ (−Ω) will
10
vanish identically at zero temperature. Physically, this cutoff corresponds
to the largest amount of energy the SET can give up to the qubit during asingle tunnel event; giving up more energy would suppress the event completely (i.e. the tunnel rates have a step-function form at zero temperature,c.f. Eq. (50)). If one included higher order processes in the tunneling (i.e.
went beyond sequential tunneling), correlated tunneling events involving
the full voltage drop over both junctions, VDS , would move this cutoff tohigher values of absolute frequency.
We now turn to the relaxation rate Γ↓ , and hence the positive frequency
parts of SQ . The calculation proceeds exactly as that for Γ↑ , the only

rjsdraft12.tex; 10/10/2002; 18:02; p.27

log[ SQ(ω) ]

28

-11

SQ(ω)+SQ(-ω) [1/Hz]

10

-2
-12

0
ω/EC

2

10

-13

10

0.01

0.1

1
Frequency (ω/EC)

10

100

Figure 5. Symmetrized SET charge noise as a function of frequency, for typical SET
parameters: g = 1, EC /kB = 2K, N = 0.33, eVDS = EC , and T = 20mK. The dashedline is the classical telegraph noise (Eq. (54)), while the dot-dashed line is the noise of
two parallel tunnel junctions (Eq. (55)). Inset: full (non-symmetrized) quantum noise for
identical SET parameters; the dashed line is the symmetric classical telegraph noise.

modification being that one now needs to include the charge states n = 2
and n = −1, as the SET could absorb enough energy from the qubit tomake transitions to these states possible. We can combine the result for Γ↓with Eq. (80) to obtain a single, compact expression for the noise at all
frequencies first obtained by Johansson et. al [14]: 13
SQ (ω) =p0 [Γ10 (ω) + Γ−1,0 (ω)] + p1 [Γ01 (ω) + Γ21 (ω)]
2
ω + 14 [Γ10 (ω) + Γ10 (−ω) + Γ01 (ω) + Γ01 (−ω)]2(81)

Shown in Figure 5 is the symmetrized noise SQ (ω) + SQ (−ω) for typical
SET parameters. One can clearly see abrupt changes in the slope of thiscurve; each of these kinks corresponds to a threshold frequency at which agiven tunneling process either turns on or turns off. For comparison, curves
13

Eq. (81) ignores additional order g/(2π) terms which arise in the denominator at
positive frequencies large enough to turn on tunneling to higher charge states; such terms
are clearly negligible in the sequential tunneling regime due to the smallness of g/(2π).

rjsdraft12.tex; 10/10/2002; 18:02; p.28

29

<P>

0.5

0.0
0.0

0.5
ngate

1.0

1.0

1.0

0.5

0.5

0.0
0.0

0.5
ngate

T1 (µs)

<n>

1.0

0.0
1.0

Figure 6. a) Average charge state of a Cooper pair box coupled to a SET, as a functionof box gate voltage, using identical SET parameters as above. The box parameters
are EC /kB = 0.5K, EJ /kB = 0.25K and the coupling constant is κ = 0.04. We also
include relaxation effects due to a 10% coupling to a 50Ω environment. The dashed curve
corresponds to assuming the SET produces classical telegraph noise, the solid curve
corresponds to using the full quantum noise of the SET, and the dashed-dot curve is thebox ground state. b) The relaxation time T1 for the same system, as a function of box
gate voltage.

corresponding to classical telegraph noise and to the uncorrelated noise of
two tunnel junctions are also shown. At low frequencies the symmetrizedtrue noise matches the classical curve; for higher frequencies, it lies above
the classical curve but below the curve corresponding to the uncorrelatedcase. The inset of this figure shows the both the negative and positivefrequency parts of SQ (ω).
It is easy to check that in the limit ω → 0, Eq. (81) recovers the classicaltelegraph expression of Eq. (54). In the high-frequency, zero temperature
limit, one can also see that Eq. (81) approaches the uncorrelated result of
Eq. (55) from below:


g 
EC
4 2π
ω 1 − 2~ω2g
→ Θ(ω)
SQ (ω) → Θ(ω)
(82)
2
g
πω
ω2 + 2 ω2πNote that at high frequencies, it is only the “direct” terms which contributeto the noise– the interference contribution is not important in the limit
of uncorrelated tunneling. The fact that the noise approaches the highfrequency limit from below results from the tendency of charging energy
induced correlations (which are present for a finite ω/EC ) to suppress
fluctuations of n, and thus suppress the noise. Note that the interpolation
between the low and high frequency limits here is very different than, e.g.,interpolating between thermal noise and zero-point fluctuation noise in atunnel junction. In the latter case, one is effectively combining two sources

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30
of noise; here, one is simply turning off correlations brought on by the
charging energy by increasing ω.
Finally, shown in Figure 6 is the average charge state of a Cooper pairbox coupled to a SET with identical parameters to that in Fig. 5. We havealso included here the relaxation effects of the environment, modelled asin Sec. 6 as a 50Ω impedance. Note that even near the box degeneracy
point, there are large deviations between the result obtained using the fullquantum noise of the SET and that obtained from using only classicaltelegraph noise. In Fig. 6b, we show the relaxation rate T1 for the same
system. Note that the differences between using the full quantum noise andthe classical expression are not so evident here.
9. Summary
In this article, we have emphasized the need to discuss quantum noise
processes using their two-sided spectral densities. Because of the quantum
nature of noise, the positive and negative frequencies are generally unequal,
in order to account for spontaneous emission. A two-level system was shown
to be an ideal spectrum analyzer for probing the quantum nature of anoise process or reservoir. With the advent of real electrical circuits which
behave as coherent two-level systems (e.g., [5],[7]), we can now build anduse quantum electrical spectrum analyzers. We also described the use ofa qubit as a theoretical tool, by following the evolution of the density
matrix of a TLS coupled to the noise-producing system of interest. This
technique appears to be quite powerful, as it can yield analytical results
for the full quantum noise spectrum of a wide variety of devices, including
the superconducting SET [17]. The distinction between the classical noise
and the quantum noise, found in this way, leads to dramatically different
predictions (c.f. Fig. 6 and Ref. [17]) for continuous measurements of qubits
with an SET. The “coupled-system” calculational approach also allows
predictions of the dephasing by the measurement, the performance relative
to the Heisenberg uncertainty limit [13], the fidelity of single-shot measurements of the qubit states, and the effects of strong coupling to the qubit.
The combined theoretical and experimental advances raise many interesting
possibilities for testing our understanding of quantum measurement theory
with mesoscopic devices.
Acknowledgements
The authors acknowledge the generous support of this work by the NSA and
ARDA under ARO contracts ARO-43387-PH-QC (RS,SG) and DAAD1902-1-044 (MD), by the NSF under DMR-0196503 & DMR-0084501 (AC,SG),

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31
the David and Lucile Packard Foundation (RS), and the W.M. Keck Foundation.

References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.

A.O. Caldeira and A.J. Leggett, Ann. Phys. 149, 374 (1983).
H.B. Callen and T.A. Welton, Phys. Rev. 83, 34 (1951).
M.H. Devoret, in Quantum Fluctuations, S. Reynaud, E. Giacobino and J. ZinnJustin, eds. (Elsevier, Amsterdam, 1995).
V. Bouchiat, D. Vion, P. Joyez, D. Esteve and M. Devoret, Physica Scripta T76,
165 (1998).
D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, and
M.H. Devoret, Science 296, 886 (2002).
J.M. Martinis et al., submitted to Phys. Rev. B, 2002.
K.W. Lehnert, K. Bladh, L.F. Spietz, D. Gunnarson, D.I. Schuster, P. Delsing, and
R.J. Schoelkopf, submitted to Phys. Rev. Lett., 2002.
V.B. Braginsky and F. Ya. Khalili, ”Quantum Measurement,” (Cambridge University Press, New York, 1992).)
A.J. Dahm et al., Phys. Rev. Lett. 22, 1416 (1969); D. Rogovin and D.J. Scalapino,Ann. Phys. 86, 1 (1974).
R.J. Schoelkopf, P.J. Burke, A.A. Kozhevnikov, D.E. Prober, and M.J. Rooks, Phys.
Rev. Lett. 78, 3370 (1997).
R. Aguado and L.P. Kouwenhoven, Phys. Rev. Lett. 84, 1986 (2000).
L. S. Levitov, H. Lee and G. B. Lesovik, J. Math. Phys. 37, 4845 (1996).
M.H. Devoret and R.J. Schoelkopf, Nature 406, 1039 (2000).
G. Johansson, Andreas Käck, and Göran Wendin, Phys. Rev. Lett. 88, 046802(2002).
H. Schoeller and G. Schön, Phys. Rev. B, 18436 (1994).
Y. Makhlin, B. Shnirman and G. Schön, Phys. Rev. Lett. 85, 4578 (2000); ibid.,
Rev. Mod. Phys. 73, 357 (2001).
A. A. Clerk, S. M. Girvin, A. K. Nguyen and A. D. Stone, Phys. Rev. Lett. 89,176804 (2002).

rjsdraft12.tex; 10/10/2002; 18:02; p.31