Quantum analysis of a linear DC SQUID mechanical displacement detector
Quantum analysis of a linear DC SQUID mechanical
displacement detector
M. P. Blencowe
Department of Physics and Astronomy,
Dartmouth College, Hanover, New Hampshire 03755
E. Buks
Department of Electrical Engineering, Technion, Haifa 32000 Israel
(Dated: October 26, 2018)
Abstract
We provide a quantum analysis of a DC SQUID mechanical displacement detector within the sub-
critical Josephson current regime. A segment of the SQUID loop forms the mechanical resonator
and motion of the latter is transduced inductively through changes in the flux threading the loop.
Expressions are derived for the detector signal response and noise, which are used to evaluate the
position and force detection sensitivity. We also investigate cooling of the mechanical resonator
due to detector back reaction.
PACS numbers: 85.25.Dq; 85.85.+j; 03.65.Ta
http://arxiv.org/abs/0704.0457v2
I. INTRODUCTION
In a series of recent experiments1,2,3 and related theoretical work,4,5,6,7,8,9 it was demon-
strated that a displacement detector based on either a normal or superconducting single
electronic transistor (SSET) can resolve the motion of a micron-scale mechanical resonator
close to the quantum limit as set by Heisenberg’s Uncertainty Principle.10,11,12 The displace-
ment transduction was achieved by capacitively coupling the gated mechanical resonator to
the SSET metallic island. When the resonator is voltage biased, motion of the latter changes
the island charging energy and hence the Cooper pair tunnel rates. The resulting modula-
tion in the source-drain tunnel current through the SSET is then read out as a signature of
the mechanical motion.
Given the success of this capacitive-based transduction method in approaching the quan-
tum limit, it is natural to consider complementary, inductive-based transduction methods
in which, for example, a superconducting quantum interference device (SQUID) is similarly
used as an intermediate quantum-limited stage between the micron-scale mechanical res-
onator and secondary amplification stages.13,14,15,16 Unavoidable, fundamental noise sources
and how they affect the SSET and SQUID devices are not necessarily the same. Furthermore,
achievable coupling strengths between each type of device and a micron-scale mechanical
resonator may be different. Therefore, it would be interesting to address the merits of the
SQUID in comparison with the established SSET for approaching the quantum limit of
displacement detection.
In the present paper, we analyze a DC SQUID-based displacement detector. The SQUID
is integrated with a mechanical resonator in the form of a doubly-clamped beam, shown
schematically in Fig. 1. Motion of the beam changes the magnetic flux Φ threading the
SQUID loop, hence modulating the current circulating the loop. We shall address the
operation of the SQUID displacement detector in the regime for which the loop current is
smaller than the Josephson junction critical current Ic and at temperatures well below the
superconducting critical temperature. We thus assume that resistive (normal) current flow
through the junctions and accompanying current noise can be neglected. (See for example
Ref. 17 for a quantum noise analysis of resistively shunted Josephson junctions and Ref. 18
for a related analysis of the DC SQUID.) Such an assumption cannot be made with the usual
mode of operation for the SSET devices, where the tunnel current unavoidably involves the
quasiparticle decay of Cooper pairs, resulting in shot noise.
As noise source, we will consider the quantum electromagnetic fluctuations within the
pump/probe feedline and also transmission line resonator that is connected to the SQUID.
This noise is a consequence of the necessary dissipative coupling to the outside world and
affects the mechanical signal output in two ways. First, the noise is added directly to the
output in the probe line and, second, the noise acts back on the mechanical resonator via
the SQUID, affecting the resonator’s motion.
With the Josephson junction plasma frequencies assumed to be much larger than the
other resonant modes of relevance for the device, the SQUID can be modeled to a good
approximation as an effective inductance that depends on the external current I entering
and exiting the loop, as well as on the applied flux. In this first of two papers, we shall make
the further approximation of neglecting the I-dependence of the SQUID effective inductance,
which requires the condition I ≪ Ic. In the sequel,19 we will relax this condition somewhat
by including the next to leading O(I2) term in the inductance and address the consequences
of this non-linear correction for quantum-limited displacement detection.
Modeling the SQUID approximately as a passive inductance element, the transmission
line resonator-mechanical resonator effective Hamiltonian is given by Eq. (24). This Hamil-
tonian describes many other detector-oscillator systems that are modeled as two coupled
harmonic oscillators, including the examples of an LC resonator capacitively coupled to a
mechanical resonator20,21 and an optical cavity coupled to a mechanically compliant mirror
via radiation pressure;22,23,24,25,26 the various systems are distinguished only by the depen-
dences of the coupling strengths on the parameters particular to each system. Thus, many
of the results of this paper are of more general relevance.
The central results of the paper are Eqs. (69) and (70), giving the detector response to a
mechanical resonator undergoing quantum Brownian motion and also subject to a classical
driving force. In the derivation of these expressions, we do not approximate the response
as a perturbation series in the coupling between the SQUID and mechanical resonator as
is conventionally done, but rather find it more natural to base our approximations instead
on assumed weak coupling between the mechanical resonator and its external heat bath
and weak classical driving force. Thus, in the context of the linear response paradigm, our
detector should properly be viewed as including the mechanical resonator degrees of freedom
as well, with the weak perturbative signal instead consisting of the heat bath force noise
and classical drive force acting on the mechanical resonator. Since the quality factors of
actual, micron-scale mechanical resonators can be very large at sub-Kelvin temperatures
(E.g., Q ∼ 105 in the experiments of Refs. 2,3), quantum electromagnetic noise in the
transmission line part of the detector can have strong back reaction effects on the motion of
the mechanical resonator, even when the coupling between the resonator and the SQUID is
very weak. One consequence that we shall consider is cooling of the mechanical resonator
fundamental mode, which requires strong back reaction damping combined with low noise.
Nevertheless, as we will also show, one can still analyze the quantum-limited detector linear
response to the mechanical resonator’s position signal using general expressions (69) and
(70), under the appropriate conditions of small pump drive and weak coupling between the
SQUID and mechanical resonator such that back reaction effects are small.
The outline of the paper is as follows. In Sec. II, we write down the SQUID-mechanical
resonator equations of motion corresponding to the circuit scheme shown in Fig. 1 and then
derive the Heisenberg equations for the various mode raising and lowering operators, subject
to the above-mentioned approximations. In Sec. III, we solve the equations within the linear
response approximation to derive the detector signal response and noise. In Sec. IV, we
analyze both the position and force detection sensitivity, and address also back reaction
cooling of the mechanical resonator. Sec. V provides concluding remarks.
II. EQUATIONS OF MOTION
A. Transmission line-SQUID-mechanical oscillator Hamiltonian
Fig. 1 shows the displacement detector scheme. The device consists of a stripline resonator
(transmission line T ) made of two sections, each of length l/2, connected via a DC SQUID
(see Refs. 27,28,29,30 for related, qubit detection schemes). The transmission line inductance
and capacitance per unit length are LT and CT respectively. The Josephson junctions in each
arm of the SQUID are assumed to have identical critical currents Ic and capacitances CJ . A
length losc segment of the SQUID loop is free to vibrate as a doubly-clamped bar resonator
and the fundamental flexural mode of interest (in the plane of the loop) is treated as a
harmonic oscillator with mass m, frequency ωm and displacement coordinate y. The total
external magnetic flux applied perpendicular to the SQUID loop is given by Φext+λBextloscy,
where Φext is the flux corresponding to the case y = 0, Bext is the normal component
of the magnetic field at the location of the vibrating loop segment (oscillator), and the
dimensionless parameter λ < 1 is a geometrical correction factor accounting for the non-
uniform displacement of the doubly-clamped resonator in the fundamental flexural mode.
p T T
FIG. 1: Scheme for the displacement detector showing the pump/probe line ‘p’, transmission line
resonator ‘T ’, and DC SQUID with mechanically compliant loop segment having effective mass m
and fundamental frequency ωm. Note that the scale of the DC SQUID is exaggerated relative to
that of the stripline for clarity.
The transmission line is weakly coupled to a pump/probe feedline (p), with inductance
and capacitance per unit length Lp and Cp respectively, employed for delivering the input and
output RF signals; the coupling can be characterized by a transmission line mode amplitude
damping rate γpT (see section IIB below). Other possible damping mechanisms in the
transmission line may be taken into account by adding a fictitious semi-infinite stripline
environment (e), weakly coupled to the transmission line characterized by mode amplitude
damping rate γeT .
31 While γeT can be made much smaller than γpT with suitable transmission
line resonator design, we shall nevertheless include both sources of damping in our analysis so
as to eventually be able to gauge their relative effects on the detector displacement sensitivity
[see Eq. (90)]. The SQUID, on the other hand, is assumed to be dissipationless. The
mechanical oscillator is also assumed to be coupled to an external heat bath (b), characterized
by mode amplitude damping rate γbm.
A convenient choice of dynamical coordinates for the SQUID are γ± = (φ1 ± φ2) /2, where
φ1 and φ2 are the gauge invariant phases across each of the two Josephson junctions.
32 For the
transmission line, we similarly use its phase field coordinate φ(x, t),30,33 where x describes
the longitudinal location along the transmission line: −l/2 < x < l/2, with the SQUID
located at x = 0. In terms of φ, the transmission line current and voltage are
IT (x, t) = −
∂φ(x, t)
VT (x, t) =
∂φ(x, t)
, (2)
where Φ0 = h/(2e) is the flux quantum. Neglecting for now the couplings to the feedline,
stripline and mechanical oscillator environments, the equations of motion for the closed
system comprising the superconducting transmission line-SQUID-mechanical oscillator are
as follows (see, e.g., Ref. 14 for a derivation of related equations of motion for a mechanical
rf-SQUID):
= (LTCT )
, (3)
ω−2J γ̈− + cos(γ+) sin(γ−) + 2β
γ− − π
(Φext + λBextloscy)
= 0, (4)
ω−2J γ̈+ + sin(γ+) cos(γ−)−
= 0, (5)
mÿ +mω2my −
λBextloscγ− = 0, (6)
where ωJ =
2πIc/(CJΦ0) is the plasma frequency of the SQUID Josephson junctions,
the dimensionless parameter βL = 2πLIc/Φ0, L is the self inductance of the SQUID, n is
an integer arising from the single-valuedness condition for the phase 2γ− around the loop,
and IT is shorthand for IT (x = 0, t). Eq. (3) is simply the wave equation for the phase field
coordinate φ(x, t) of the transmission line. Eq. (4) describes the current circulating the loop,
which depends on the external flux threading the loop. Eq. (5) describes the average current
threading the loop, which from current conservation is equal to one-half the transmission
line current at x = 0. With the circulating SQUID current given by Φ0γ−/(πL) (up to a
Φext dependent term), we recognize in Eq. (6) the Lorentz force acting on the mechanical
oscillator.
In addition to the equations of motion, we have the following current and voltage bound-
ary conditions:
IT (x = ±l/2, t) = 0 (7)
∂ (Leff [Φext(y), IT ]IT )
= VT (0
−, t)− VT (0+, t), (8)
where the external flux and current-dependent, effective inductance Leff [Φext(y), IT ] of the
SQUID as ‘seen’ by the transmission line is
Leff [Φext(y), IT ] =
, (9)
with Φext(y) = Φext + λBextloscy. Note that we have set n = 0, since observable quantities
do not depend on n.
We now make the following assumptions and consequent approximations: (a) ωJ ≫ ωT ≫
ωm (where ωT is the relevant resonant mode of the transmission line); neglect the SQUID
inertia terms ω−2J γ̈±. (b) βL ≪ 1; solve for γ± as series expansions to first order in βL.
(c) |Bextloscy| /Φ0 ≪ 1; series expand the equations of motion to first order in y(t). (d)
|IT/Ic| =
2πLT Ic
∂φ(0,t)
≪ 1; series expand the equations of motion to second order in IT .
With ωJ ’s typically in the tens of GHz, assumption (a) is reasonable. From Eq. (4), we
see that a small βL value prevents the γ− coordinate from getting trapped in its various
potential minima, causing unwanted hysteresis. With the γ+ expansion in IT consisting of
only odd powers, approximations (a) and (d) amount to describing the SQUID simply as a
current independent, Φext-tunable passive inductance element Leff [Φext(y)] that also depends
on the mechanical oscillator position coordinate y. Including the next-to-leading, I3T term in
the γ+ expansion gives an I
T -dependent, nonlinear correction to the SQUID effective induc-
tance. The consequences of including this nonlinear correction term for the quantum-limited
displacement detection sensitivity will be considered in a forthcoming paper.19 Solving for
γ+ to order IT and substituting in Eq. (9), we obtain:
Leff [Φext(y)] ≈
πΦext(y)
, (10)
where the self inductance L contribution has been neglected since it is of order βL ≪ 1.
Solving for γ− to order I
T and substituting into Eq. (6), we obtain for the mechanical
oscillator equation of motion:
mÿ +mω2my −
πλBextloscI
tan (πΦext/Φ0) sec (πΦext/Φ0) = 0, (11)
where from (c), we have set y = 0 in the solution for γ− and have dropped an overall
constant term. Since the γ− expansion in IT consists only of even powers, we must go to
second order in IT so as to have a non-trivial transmission line-oscillator effective coupling.
Thus, the SQUID phase coordinates γ± have been completely eliminated from the equations
of motion, a consequence of approximation (a); the SQUID mediates the interaction between
the mechanical oscillator coordinate y and transmission line coordinate φ without retardation
effects.
From Eq. (11), it might appear that the force on the mechanical oscillator due to the
transmission line can be made arbitrarily large by tuning Φext close to Φ0/2. Note, however,
that the proper conditions for the validity of the IT and βL expansions are:
sec (πΦext/Φ0)
≪ 1 (12)
|βL sec (πΦext/Φ0)| ≪ 1. (13)
We now restrict ourselves to a single transmission line mode and derive approximate
equations of motion for the mode amplitude. Suppose that the mechanical oscillator position
coordinate is held fixed at y = 0. The following phase field satisfies the current boundary
conditions (7):
φ(x, t) =
−φ(t) cos [k0 (x+ l/2)] ; x < 0
+φ(t) cos [k0 (x− l/2)] ; x > 0
, (14)
with the wavenumber k0 determined by the voltage boundary condition (8):
Leff (Φext)
. (15)
The wave equation (3) gives for the transmission mode frequency: ωT = k0/
LTCT . Substi-
tuting the phase field (14) into the IT part of the oscillator equation of motion (11) further-
more gives the transmission line force acting on the oscillator with fixed coordinate y = 0.
Now release the mechanical oscillator coordinate and suppose that for small [condition (c)] ,
slow [condition (a)] displacements, the force is the same to a good approximation.Then the
oscillator equation of motion becomes
mÿ(t) +mω2my(t) +
sin2 (k0l/2)
−λBextlosc
(Φ0/2π)
4πLT lIc
tan (πΦext/Φ0) sec (πΦext/Φ0)
2(t) = 0, (16)
From Eq. (16), we can determine the mechanical sector of the Lagrangian, along with the
interaction potential involving y and the mode amplitude φ. The remaining transmission
line sector follows from the wave equation (3) and we thus have for the total Lagrangian:
φ, y, φ̇, ẏ
mẏ2 − 1
mω2my
sin2 (k0l/2)
φ̇2 − 1
1− λBextloscy
(Φ0/2π)
4πLT lIc
tan (πΦext/Φ0) sec (πΦext/Φ0)
. (17)
From Eq. (17), we see that for motion occuring on the much longer timescale ω−1m ≫ ω−1T ,
the mechanical oscillator has the effect of modulating the frequency of the transmission line
mode.
The associated Hamiltonian is
H (φ, y, pφ, py) =
sin2 (k0l/2)
p2φ +
sin2 (k0l/2)
1− λBextloscy
(Φ0/2π)
4πLT lIc
tan (πΦext/Φ0) sec (πΦext/Φ0)
mω2my
2. (18)
Let us now quantize. For the transmission line mode coordinate, the raising(lowering)
operator is defined as:
â±T =
CT l (Φ0/2π)
sin2 (k0l/2)
sin2 (k0l/2)ωT φ̂∓ ip̂φ
and for the mechanical oscillator
â±m =
(mωŷ ∓ ip̂y) . (20)
In terms of these operators, the Hamiltionian (18) becomes (for notational convenience we
omit from now on the ‘hats’ on the operators and also the ‘minus’ superscript on the lowering
operator):
H = ~ωTa
T aT + ~ωma
mam +
~ωTKTm
aT + a
am + a
, (21)
where the dimensionless coupling parameter between the mechanical oscillator and trans-
mission line mode is
KTm = −
λBextlosc∆xzp
(Φ0/2π)
4πLT lIc
tan (πΦext/Φ0) sec (πΦext/Φ0) , (22)
with ∆xzp =
~/(2mωm) the zero-point uncertainty of the mechanical oscillator. From
expression (10) for the effective inductance, another way to express the coupling parameter
is as follows:
KTm = −
λBextlosc∆xzp
(Φ0/2π)
dLeff/dΦext
. (23)
From Eq. (23), we see that in order to increase the coupling between the mechanical oscillator
and transmission line, the SQUID effective inductance-to-transmission line inductance ratio
must be increased. The advantage of using a SQUID over an ordinary, geometrical mutual
inductance between a transmission line and micron-sized mechanical oscillator is that the
former can give a much larger effective inductance. As we shall see in Sec. IV, just requiring
that the inductances be matched such that Φ0
dLeff/dΦext
∼ 1 is sufficient for strong back
reaction effects with modest drive powers, even though the other term in KTm describing
the flux induced for a zero-point displacement is typically very small.
Assuming then that KTm ≪ 1 and making the rotating wave approximation (RWA) for
the ‘T ’ part of the interaction term in the system Hamiltonian (21), i.e., neglecting the terms
(aT )
2 and (a+T )
2, we have (up to an unimportant additive constant):
H = ~ωTa
T aT + ~ωma
mam + ~ωTKTma
am + a
. (24)
Many other systems are modeled by this form of Hamiltonian, a notable example being
the single mode of an optical cavity interacting via radiation pressure with a mechanically
compliant mirror.22,23,24,25,26 Thus, much of the subsequent analysis will be relevant to a
broad class of coupled resonator devices–not to just the transmission line-SQUID-mechanical
resonator system.
B. Open system Heisenberg equations of motion
So far, we have treated the transmission line and mechanical resonator as a closed system
with SQUID-induced effective coupling . Of course, a real transmission line mode will expe-
rience damping and accompanying fluctuations, not least because it must be coupled to the
outside world in order for its state to be measured. Furthermore, the mechanical resonator
mode will of course be damped even when decoupled from the SQUID. It is straightforward
to incorporate the various baths and pump/probe feedline in terms of raising/lowering op-
erators. Assuming weak system-bath couplings, which again justify the RWA, we have for
the full Hamiltonian:
H = ~ωTa
T aT + ~ωma
mam + ~ωTKTma
am + a
dωωa+p (ω)ap(ω) + ~
dωωa+e (ω)ae(ω) + ~
dωωa+b (ω)ab(ω)
K∗pTa
p (ω)aT +KpTa
T ap(ω)
K∗eTa
e (ω)aT +KpTa
T ae(ω)
K∗bma
b (ω)am +Kbma
mab(ω)
(am + a
m)Fext(t), (25)
where ap denotes the pump/probe (p) feed line operator, ae the transmission line bath (‘e’ for
‘environment’) operator, and ab the mechanical resonator bath (b) operator. These operators
satisfy the usual canonical commutation relations:
ai(ω), a
= δijδ(ω − ω′). (26)
The couplings between these baths and the transmission line and mechanical resonator
systems are denoted as KpT , KeT , and Kbm. Note we have also included for generality
a classical driving force Fext(t) acting on the mechanical resonator. This allows us the
opportunity to later on analyze quantum limits on force detection in addition to displacement
detection.
Within the RWA, it is straightforward to solve the Heisenberg equations for the bath
operators and substitute these solutions into the Heisenberg equations for the transmission
line and mechanical oscillator to give
= −iωmam +
Fext(t)− iωTKTma+T aT
dω |KTm|2
dt′e−iω(t−t
′)am(t
′)− i
dωKbme
−iω(t−t0)ab(ω, t0) (27)
= −iωTaT − iωTKTmaT
am + a
dω |KpT |2
dt′e−iω(t−t
′)aT (t
′)− i
dωKpTe
−iω(t−t0)ap(ω, t0)
dω |KeT |2
dt′e−iω(t−t
′)aT (t
′)− i
dωKeTe
−iω(t−t0)ae(ω, t0). (28)
We now make the so-called ‘first Markov approximation’,34,35 in which the frequency
dependences of the couplings to the baths are neglected:
KpT (ω) =
eiφpT
KeT (ω) =
eiφeT
Kbm(ω) =
eiφbm , (29)
where the γ’s and φ’s are independent of ω as stated. The Heisenberg equations of motion
(27) and (28) then simplify to
= −iωmam +
Fext(t)− iωTKTma+T aT
−γbmam(t)− i
2γbme
iφbmainb (t) (30)
= −iωTaT − iωTKTmaT
am + a
−γpTaT (t)− i
2γpTe
iφpT ainp (t)
−γeTaT (t)− i
2γeTe
iφeT aine (t), (31)
where the γi’s are the various mode amplitude damping rates (assumed much smaller than
their associated mode frequencies) and the ‘in’ operators10,31,34,35 are defined as
aini (t) =
dωe−iω(t−t0)ai(ω, t0), (32)
with t > t0. The time t0 can be taken to be an instant in the distant past before the
measurement commences and when the initial conditions are specified (see below). We can
similarly define ‘out’ operators:
aouti (t) =
dωe−iω(t−t1)ai(ω, t1), (33)
with t1 > t. The time t1 can be taken to be an instant in the distant future after the measure-
ment has finished. From the Heisenberg equations for the bath operators and the definitions
of the ‘in’ and ‘out’ operators, we obtain the following identities between them:34,35
aoutp (t)− ainp (t) = −i
2γpTe
−iφpT aT (t)
aoutb (t)− ainb (t) = −i
2γbme
−iφbmam(t)
aoute (t)− aine (t) = −i
2γeTe
−iφeT aT (t). (34)
In outline, the method of solution runs in principle as follows:31,34,35,36 (1) specify the ‘in’
operators. (2) Solve for the system operators am(t) and aT (t) in terms of the ‘in’ operators.
(3) Use the relevant identity (34) to determine the ‘out’ operator aoutp (t), which yields the
desired probe signal. It is more convenient to solve the Heisenberg equations in the frequency
domain with the Fourier transformed operators O(t) = 1√
−∞ dωe
−iωtO(ω). The equations
for the system operators then become
am(ω) =
ω − ωm + iγbm
2γbme
iφbmainb (ω)−
2m~ωm
Fext(ω)
ωTKTm
aT (ω
′)a+T (ω
′ − ω) + a+T (ω
′)aT (ω + ω
aT (ω) =
ω − ωT + i(γpT + γeT )
2γpTe
iφpT ainp (ω) +
2γeTe
iφeT aine (ω)
ωTKTm√
dω′aT (ω
am(ω − ω′) + a+m(ω′ − ω)
, (36)
while the relevant ‘in/out’ operator identity becomes
aoutp (ω) = −i
2γpTe
−iφpT aT (ω) + a
p (ω). (37)
C. Observables and ‘in’ states
Before proceeding with the solution to Eqs. (35) and (36), let us first devote some time
to deriving expressions for observables that we actually measure in terms of aoutp (ω). Model
the pump/probe feedline as a semi-infinite transmission line −∞ < x < 0. Solving the wave
equation for the decoupled transmission line and then using the expressions (1), (2) relating
the current/voltage to the phase coordinate, we obtain
Iout(x, t) = −
sin (ωx/vp)
e−iωtaoutp (ω) + e
iωtaout+p (ω)
V out(x, t) = i
cos (ωx/vp)
e−iωtaoutp (ω)− eiωtaout+p (ω)
, (39)
where the sinusoidal x dependence in the current expression follows from the vanishing of the
current boundary condition at x = 0, the feedline impedance is Zp =
Lp/Cp and the wave
propagation velocity is vp = 1/
LpCp. Suppose the current/volt meter is at x → −∞, so
that the actual observables correspond to measuring the left-propagating component of the
current/voltage. Then decomposing the x-dependent trig terms into their real and imaginary
parts, we can identify the left propagating current/voltage operators as
Iout(x, t) = −i
e−iω(x/vp+t)
aoutp (ω)− aout+p (−ω)
+eiω(x/vp+t)
aoutp (−ω)− aout+p (ω)
V out(x, t) = i
e−iω(x/vp+t)
aoutp (ω)− aout+p (−ω)
+eiω(x/vp+t)
aoutp (−ω)− aout+p (ω)
. (41)
The output signal of interest due to the mechanical oscillator signal input will lie within
some bandwidth δω centered at ωs, the ‘signal’ frequency, and so we define the filtered
output current Iout (x, t|ωs, δω) and voltage V out (x, t|ωs, δω) to be the same as the above,
left-moving operators, but with the integration range instead restricted to the interval
[ωs − δω/2, ωs + δω/2].
Since the motion of the mechanical resonator modulates the transmission line frequency,
one way to transduce displacements is to measure the relative phase shift between the ‘in’
pump current and ‘out’ probe current using the homodyne detection procedure.35 Another
common way is to measure the ‘out’ power relative to the ‘in’ power, or equivalently the
mean-squared current/voltage (all three quantities differ by trivial factors of Zp). We will
discuss the latter method of transduction; the former, homodyne method can be straight-
forwardly addressed using similar techniques to those presented here. Thus, we consider the
following expectation value:
δIout (x, t|ωs, δω)
Iout (x, t|ωs, δω)
Iout (x, t|ωs, δω)
, (42)
where the angle brackets denote an ensemble average with respect to the ‘in’ states of
the various baths and feedline (see below). If the mechanical oscillator is being driven
by a classical external force whose fluctuations are invariant under time translations, i.e.,
〈Fext(t)Fext(t′)〉 = C(t− t′), then the above, mean-squared current will be time-independent.
Alternatively, if Fext(t) is, e.g., some deterministic, AC drive, then we must also time-average
so as to get a time-independent measure of the detector response:
[δIout (x, t|ωs, δω)]2
∫ TM/2
−TM/2
Iout (x, t|ωs, δω)
, (43)
where TM is duration of the measurement, assumed much larger than all other timescales
associated with the detector dynamics. We have also assumed that the time-averaged cur-
rent vanishes in the signal bandwidth of interest: 〈Iout (ωs, δω)〉 = 0. Substituting in the
expression (40) for Iout (x, t|ωs, δω) in terms of the aoutp operators, we obtain after some
algebra:
[δIout (ωs, δω)]
∫ ωs+δω/2
ωs−δω/2
dω1dω2
(ω1 − ω2) TM
sin [(ω1 − ω2)TM/2]
aoutp (ω1)a
p (ω2) + a
p (ω2)a
p (ω1)
. (44)
As ‘in’ states, we suppose kBT ≪ ~ωT , such that the relevant transmission line ‘in’ bath
modes (ωe ∼ ωT ) are assumed to be approximately in the vacuum state. On the other hand,
with the mechanical mode typically at a much lower frequency ωm ≪ ωT , we assume that its
relevant ‘in’ bath modes (ωb ∼ ωm) are in the proper, non-zero temperature thermal state.
For the pump/probe feedline, we consider the following coherent state:30
|{α(ω)}〉p = exp
dωα(ω)
ain+p (ω)− ainp (ω)
|0〉p , (45)
where |0〉p is the vacuum state and
α(ω) = −I0
e−(ω−ωp)
, (46)
normalized such that the amplitude of the expectation value of I in [the right propagating
version of (40) with aoutp replaced by a
p ] with respect to this state is just I0. Again, we
suppose kBT ≪ ~ωp, so that thermal fluctuations of the feedline are neglected. The fre-
quency width of this pump drive is assumed to be the inverse lifetime of the measurement.
Below we shall see that the output mechanical signal will appear as two ‘satellite’ peaks on
either side of the central peak at ωp due to the pump signal, i.e, the mechanical signal can
be extracted by centering the filter at either of ωs = ωp ± ωm (up to a renormalization of
the mechanical oscillator frequency), corresponding to the anti-Stokes and Stokes bands.
Note that we do not have to specify the initial t0 states of the mechanical resonator and
transmission line systems; aT (t0) and am(t0)-dependent initial transients have been dropped
in the above equations for aT (ω) and am(ω), since they give a negligible contribution to the
long-time, steady-state behavior of interest.
III. SOLVING THE EQUATIONS OF MOTION
A. Linear response approximation
We are now ready to solve for
[δIout]
. Introduce the following shorthand notation:
ST (ω) =
2γpTe
iφpT ainp (ω) +
2γeTe
iφeT aine (ω)
Sm(ω) =
2γbme
iφbmainb (ω)−
2m~ωm
Fext(ω)
ωTKTm√
, (47)
and γT = γpT +γeT , the net transmission line mode amplitude dissipation rate due to loss via
the probe line and the transmission line bath. Substituting Eq. (35) for am(ω) into Eq. (36)
for aT (ω) yields the following, single equation in terms of aT (ω) only:
aT (ω) =
dω′aT (ω − ω′)A(ω, ω′) +
dω′B(ω, ω′)aT (ω − ω′)
aT (ω
′′)a+T (ω
′′ − ω′) + a+T (ω
′′)aT (ω
′′ + ω′)
+ C(ω), (48)
where, for the convenience of subsequent calculations, we have made this equation as concise
as possible with the following definitions:
A(ω, ω′) =
ω − ωT + iγT
ω′ − ωm + iγbm
S+m(−ω′)
−ω′ − ωm − iγbm
B(ω, ω′) =
ω − ωT + iγT
ω′ − ωm + iγbm
−ω′ − ωm − iγbm
C(ω) =
ST (ω)
ω − ωT + iγT
. (49)
We expand Eq. (48) for aT (ω) to first order in the mechanical oscillator bath operator
ainb (ω) and external driving force Fext(ω) [equivalently expand in A(ω, ω
′)]: aT (ω) ≈ a(0)T (ω)+
T (ω), where
T (ω) =
dω′B(ω, ω′)a
T (ω − ω
′′ − ω′) + a(0)+T (ω
′′ + ω′)
+ C(ω) (50)
T (ω) =
T (ω − ω
′)A(ω, ω′) +
dω′B(ω, ω′)a
T (ω − ω
′′ − ω′) + a(0)+T (ω
′′ + ω′)
dω′B(ω, ω′)a
T (ω − ω
′′ − ω′)
′′ + ω′) + a
′′ − ω′)
′′ + ω′)
. (51)
Eq. (50) then yields the detector noise, while (51) yields the detector response to the signal
within the linear response approximation. Thus, our approach here is to treat the mechanical
oscillator as part of the detector degrees of freedom, with the signal defined as the thermal
bath fluctuations and classical external force acting on the oscillator. This is the appropriate
viewpoint for force detection. On the other hand, if the focus is on measuring the quantum
state of the mechanical oscillator itself, then the oscillator should not be included as part
of the detector degrees of freedom. Nevertheless, as we shall later see, the latter viewpoint
can be straightforwardly extracted from the former under not too strong coupling KTm and
pump drive current amplitude I0 conditions.
B. Semiclassical approximation
The sequence of solution steps to Eqs. (50) and (51) are in principle as follows: (1) Solve
first equation (50) for a
T (ω) in terms of B(ω, ω
′) and C(ω); (2) Substitute the solution for
T (ω) into Eq. (51) for a
T (ω) and invert this Eq. (which is linear in a
T (ω)) to obtain
the solution for a
T (ω) in terms of A(ω, ω
′), B(ω, ω′), and C(ω). It is not clear how to
carry out these steps in practice, however, since the equations involve products of non-
commuting operators. Thus, we must find some way to solve by further approximation.
The key observation is that the feedline is in a coherent state, which is classical-like for
sufficiently large current amplitude I0 so as to ensure signal amplification. We therefore
decompose a
T (ω) into a classical, expectation-valued part and quantum, operator-valued
fluctuation part, a
T (ω) =
T (ω)
T (ω), and subtitute into Eq. (50) for a
T (ω),
linearizing with respect to the quantum fluctuation δa
T (ω). This gives two equations, one
for the expectation value
T (ω)
dω′B(ω, ω′)
T (ω − ω
′′ − ω′)
′′ + ω′)
+ 〈C(ω)〉 (52)
and the other for the quantum fluctuation:
T (ω) =
dω′B(ω, ω′)δa
T (ω − ω
′′ − ω′)
′′ + ω′)
dω′B(ω, ω′)
T (ω − ω
′′ − ω′)
′′ − ω′)
′′ + ω′)
′′ + ω′)
+ δC(ω). (53)
Eq. (51) for a
T (ω) is approximated by replacing a
T (ω) with its expectation value
T (ω)
i.e., we drop the quantum fluctuation part δa
T (ω). This is because Eq. (51) already depends
linearly on the quantum fluctuating signal term A(ω, ω′), which we of course want to keep.
Dropping the δa
T (ω) contribution to Eq. (51) amounts to neglecting multiplicative detector
noise, which is reasonable given that we are concerned with large signal amplification.
C. Complete solution to detector signal response and noise
The sequence of solutions steps are therefore in practice as follows: (1) Solve Eq. (52) first
T (ω)
; (2) Substitute this solution into Eq. (51) for a
T (ω) and invert; (3) Substitute
the solution for
T (ω)
into the Eq. (53) for δa
T (ω) and invert; (4) Use these solutions for
T (ω) and δa
T (ω) to determine the detector signal and noise terms, respectively. Beginning
with step (1), we have
〈C(ω)〉 = −
2γpTe
γT − i∆ω
ainp (ω)
2γpTe
γT − i∆ω
e−(ω−ωp)
/2, (54)
where ∆ω = ωp−ωT is the detuning frequency (not to be confused with the bandwidth δω)
and note
aine (ω)
= 0 (recall, we assume the transmission line resonant frequency ωT mode
is in the vacuum state). Given that TM is the longest timescale in the system dynamics,
〈C(ω)〉 is sharply peaked about the frequency ωp and we will therefore approximate the
exponential with a delta function: 〈C(ω)〉 = cδ(ω − ωp), where
2πeiφpT
γT − i∆ω
I20ZpγpT
. (55)
Considering for the moment an iterative solution to Eq. (52) for 〈a(0)T (ω)〉, we see that
〈a(0)T (ω)〉 must also have the form of a delta function peaked at ωp: 〈a
T (ω)〉 = χδ(ω − ωp).
Substituting this ansatz into Eq. (52), we obtain the following equation for χ:
χ = 2χ |χ|2B(ωp, 0) + c. (56)
This equation has a rather involved analytical solution. For sufficiently large |c|2 |B(ωp, 0)|
the response can become bistable (i.e., two locally stable solutions for χ). This region will
not be discussed in the present paper, however. When we consider actual device parameters
later in Sec. IV, we will assume sufficiently small drive such that χ ≈ c, allowing much
simpler analytical expressions to be written down for the detector response.
Proceeding now to step (2), we substitute the expectation value
T (ω)
= χδ(ω − ωp)
for the operator a
T (ω) into Eq. (51) for a
T (ω). Carrying out the integrals, we obtain
1− 2 |χ|2 [B(ω, 0) +B(ω, ω − ωp)]
T (ω)− 2χ
2B(ω, ω − ωp)a(1)+T (2ωp − ω)
= χA(ω, ω − ωp). (57)
Before we can invert to obtain a
T (ω), we require a second linearly independent equation
also involving a
T (2ωp−ω) and a
T (ω). This equation can be obtained by replacing ω with
2ωp − ω in Eq. (57) and then taking the adjoint:
1 + 2 |χ|2 [B(ω − 2∆ω, 0) +B(ω − 2∆ω, ω − ωp)]
T (2ωp − ω)
+2χ∗2B(ω − 2∆ω, ω − ωp)a(1)T (ω) = −χ
∗A(ω − 2∆ω, ω − ωp), (58)
where we have used the identities A+(2ωp − ω, ωp − ω) = −A(ω − ∆ω, ω − ωp), B∗(2ωp −
ω, ωp − ω) = −B(ω − 2∆ω, ω − ωp), and B∗(2ωp − ω, 0) = −B(ω − 2∆ω, 0). Inverting, we
obtain
T (ω) = α1(ω)A(ω, ω − ωp) + α2(ω)A(ω − 2∆ω, ω − ωp), (59)
where
α1(ω) = D(ω)
−1{1 + 2 |χ|2 [B(ω − 2∆ω, 0) +B(ω − 2∆ω, ω − ωp)]
χ (60)
α2(ω) = −2D(ω)−1 |χ|2B(ω, ω − ωp)χ, (61)
with determinant
D(ω) =
1− 2 |χ|2 [B(ω, 0) +B(ω, ω − ωp)]
1 + 2 |χ|2 [B(ω − 2∆ω, 0) +B(ω − 2∆ω, ω − ωp)]
+4 |χ|4B(ω, ω − ωp)B(ω − 2∆ω, ω − ωp). (62)
Moving on now to step (3), we substitute the expectation value 〈a(0)T (ω)〉 = χδ(ω − ωp)
into Eq. (53) for δa
T (ω) and carry out the integrals to obtain:
1− 2 |χ|2 [B(ω, 0) +B(ω, ω − ωp)]
T (ω)− 2χ
2B(ω, ω − ωp)δa(0)+T (2ωp − ω)
= δC(ω). (63)
Replacing ω with 2ωp − ω in Eq. (63) and then taking the adjoint:
1 + 2 |χ|2 [B(ω − 2∆ω, 0) +B(ω − 2∆ω, ω − ωp)]
T (2ωp − ω)
+2χ∗2B(ω − 2∆ω, ω − ωp)δa(0)T (ω) = δC
+(2ωp − ω). (64)
Inverting Eqs. (63) and (64), we obtain
T (ω) = β1(ω)δC(ω) + β2(ω)δC
+(2ωp − ω), (65)
where
β1(ω) = D(ω)
−1{1 + 2 |χ|2 [B(ω − 2∆ω, 0) +B(ω − 2∆ω, ω − ωp)]
β2(ω) = 2D(ω)
−1χ2B(ω, ω − ωp) (67)
We are now ready to carry out step (4). To obtain the detector response, we substi-
tute into expression (44) for
[δIout]
the linear response approximation to the ‘out’ probe
operator [see Eq. (37)]:
aoutp (ω) =
2γpTe
−iφpT a
T (ω)
2γpTe
−iφpT δa
T (ω) + δa
p (ω)
. (68)
The first square-bracketed term will give the signal contribution to the detector response,
while the second bracketed term gives the noise contribution. Note that the average values
T (ω)
ainp (ω)
are not required in the noise term since they give negligible contri-
bution in the signal bandwidths of interest centered at ωs = ωp ± ωm. Substituting in the
signal part of aoutp (ω), we obtain after some algebra:
[δIout (ωs, δω)]
signal
I0KTmωT
)2 γ2pT
γ2T +∆ω
∫ ωs+δω/2
ωs−δω/2
(ω − ωp +∆ω)2 + γ2T
α1(ω)
α2(ω)
ω − ωp +∆ω + iγT
ω − ωp −∆ω + iγT
(ω − ωp − ωm)2 + γ2bm
[2n(ω − ωp) + 1] +
(ωp − ω − ωm)2 + γ2bm
[2n(ωp − ω) + 1]
I0KTmωT
)2 γ2pT
γ2T +∆ω
2m~ωmγbm
∫ ωs+δω/2
ωs−δω/2
dωdω′
(ω − ωp +∆ω)2 + γ2T
α1(ω)
α2(ω)
ω − ωp +∆ω + iγT
ω − ωp −∆ω + iγT
×sin [(ω − ω
′)TM/2]
(ω − ω′) TM/2
(ω − ωp − ωm)2 + γ2bm
Fext(ω − ωp)F ∗ext(ω′ − ωp)
(ωp − ω − ωm)2 + γ2bm
Fext(ωp − ω)F ∗ext(ωp − ω′)
, (69)
where n(ω) =
e~ω/kBT − 1
is the Bose-Einstein thermal occupation number average for
bath mode ω. The signal part of the detector response comprises a thermal component
and a classical force component. In the limit of weak coupling KTm → 0 and or small
drive current amplitude I0 → 0, we have α1(ω)/c → 1, α2(ω)/c → 0 and we note that the
frequency resolved detector response has the form of two Lorentzians centered at ωp ± ωm.
The resulting expression for the detector response coincides with an O(K2Tm) perturbative
solution to the detector response (44) via the linear response Eqs. (50) and (51) (but no
semiclassical approximation). However, as shall be described in Sec. IV, when the current
drive is not small and or coupling is not weak, then the αi terms will modify this simple
form, at the next level of approximation renormalizing the Lorentzians, i.e., shifting their
location and changing their width.
Substituting in the noise part of aoutp (ω), we obtain after some algebra:
[δIout (ωs, δω)]
noise
= Z−1p
∫ ωs+δω/2
ωs−δω/2
2γTγpT
(ω − ωp +∆ω)2 + γ2T
|β1(ω)|2 +
(ω − ωp +∆ω)2 + γ2T
(ω − ωp −∆ω)2 + γ2T
|β2(ω)|2 − Re [β1(ω)] +
(ω − ωp +∆ω)
Im [β1(ω)]
+Z−1p
. (70)
The noise part of the detector response comprises a back reaction component (the integral
term) where transmission line noise drives the mechanical oscillator via the SQUID coupling,
and a component that is added at the output due to zero-point fluctuations in the probe
line. While not as obvious given the form of Eq. (70), one may again verify (see Sec. IV) that
the detector back reaction on the mechanical oscillator takes the form of two Lorentzians
centered at ωp ± ωm in the weak coupling and or weak current drive limit, coinciding with
an O(K2Tm) perturbative calculation.
Eqs. (69) and (70) are the main results of the paper, their sum giving the net output
mean-squared current.
D. Quantum bound on noise
As articulated by Caves,10 the fact that the ‘in’ and ‘out’ operators satisfy canonical
commutation relations places a lower, quantum limit on the noise contribution to the detector
response, Eq. (70). We now derive this quantum limit. First write the ‘out’ operator (68)
aoutp (ω) = −i
2γpTe
−iφpT a
T (ω) +N(ω), (71)
where N(ω) = −i
2γpTe
−iφpT δa
T (ω) + δa
p (ω) is the noise part. Taking commutators, we
have the following identity relating the noise and signal operator terms:
N(ω), N+(ω′)
= δ(ω − ω′)− 2γpT
T (ω), a
. (72)
Now, from the Heisenberg Uncertainty Principle, one can derive the following general in-
equality:
N [f ]N+[f ] +N+[f ]N [f ]
N [f ], N+[f ]
∣ , (73)
where N [f ] =
dωf(ω)N(ω) and f(ω) is an arbitrary function. Inserting the commutator
identity (72), Eq. (73) becomes
N [f ]N+[f ] +N+[f ]N [f ]
dω |f(ω)|2 − 2γpT
T [f ], a
T [f ]
. (74)
Choosing the ‘filter’ function f(ω) = ωΘ(ω − ωs + δω/2)Θ(ωs + δω/2 − ω) and evaluating
the commutator, we obtain the following lower bound on the detector noise:
[δIout (ωs, δω)]
noise
I0KTmωT
)2 γ2pT
γ2T +∆ω
∫ ωs+δω/2
ωs−δω/2
(ω − ωp +∆ω)2 + γ2T
α1(ω)
α2(ω)
ω − ωp +∆ω + iγT
ω − ωp −∆ω + iγT
(ω − ωp − ωm)2 + γ2bm
− 2γbm
(ωp − ω − ωm)2 + γ2bm
. (75)
In the next section we will address the extent to which the detector noise can approach the
quantum bound on the right hand side of Eq. (75), depending on the current drive amplitude
I0 and other detector parameters.
IV. RESULTS
A. Analytical approximations
To gain a better understanding of the detector response, we now provide analytical ap-
proximations to Eqs. (69) and (70) that are valid under the condition |c|2 |B(ωp, 0)| ≪ 1
such that χ ≈ c [see Eq. (56)], i.e., the expectation value
T (ω)
for the transmission line
depends approximately only on the pump/probe feedline state and not on the mechanical
oscillator state. Explicitly, this condition reads:
2I20ZpK
TmωTγpT
~ωm (γ
T +∆ω
≪ 1, (76)
placing an upper limit on I0 and KTm for the validity of this approximation. We also
assume that the mechanical and transmission line mode frequencies are widely separated:
ωm ≪ ωT , and with small damping rates: γbm ≪ ωm, γT ≪ ωT . We do not restrict
the relative magnitudes of ωm and γT , however. A simple picture emerges in which the
detector back reaction ‘renormalizes’ the mechanical oscillator frequency and damping rate:
ωm → Rωωm and γbm → Rγγbm, where
Rωωm = ωm +
|c|2ω2TK2Tm
|c|2ω2TK2Tm [γ2T +∆ω2 − ω2m]
γ2T + (∆ω + ωm)
γ2T + (∆ω − ωm)
] (77)
Rγγbm = γbm −
|c|2ω2TK2Tm
2|c|2ω2TK2TmωmγT
γ2T + (∆ω + ωm)
γ2T + (∆ω − ωm)
] , (78)
where c is defined in Eq. (55). With the measurement filter bandwidth centered at either of
ωs = ωp±Rωωm, the approximation to Eq. (69) for the signal response is (with the classical
force term omitted):
[δIout (ωs = ωp ±Rωωm, δω)]2
signal
I0KTmωT
)2 γ2pT
γ2T +∆ω
γ2T + (∆ω ± ωm)
∫ ωs+δω/2
ωs−δω/2
(ω − ωp ∓Rωωm)2 + (Rγγbm)2
[2n(Rωωm) + 1] . (79)
When there is a classical force acting on the mechanical oscillator, we must add to Eq. (79)
the term
I0KTmωT
)2 γ2pT
γ2T +∆ω
2m~ωmγbm
∫ ωs+δω/2
ωs−δω/2
dωdω′
(ω − ωp +∆ω)2 + γ2T
×sin [(ω − ω
′)TM/2]
(ω − ω′)TM/2
(ω − ωp − Rωωm)2 + (Rγγbm)2
Fext(ω − ωp)F ∗ext(ω′ − ωp)
(ωp − ω − Rωωm)2 + (Rγγbm)2
Fext(ωp − ω)F ∗ext(ωp − ω′)
. (80)
The approximation to Eq. (70) for the detector noise is
[δIout (ωs = ωp ±Rωωm, δω)]2
noise
I0KTmωT
)2 γ2pT
γ2T +∆ω
γ2T + (∆ω ± ωm)
∫ ωs+δω/2
ωs−δω/2
(ω − ωp ∓ Rωωm)2 + (Rγγbm)2
N± + Z
, (81)
where the back reaction noise parameter is
|c|2K2Tmω2TγT
γ2T + (∆ω ∓ ωm)
] ∓ 1 = 2I
TmωTγTγpT
~γbm [γ
T +∆ω
γ2T + (∆ω ∓ ωm)
] ∓ 1. (82)
The ∓1 term in the back reaction noise parameter depends on whether the filter is centered
at ωs = ωp + ωm or ωs = ωp − ωm and corresponds respectively to ‘phase preserving’ or
‘phase conjugating’ detection as discussed in Caves.10 In the limit I0 → 0 and or KTm → 0,
we see from Eqs. (79), (81), and (82) that the back reaction noise amounts to doubling the
oscillator quantum zero-point motion signal in the phase conjugating case, while the back
reaction noise exactly cancels the quantum zero-point motion signal in the phase preserving
case. In both cases, the noise coincides with the lower quantum bound (75). However, in this
small drive/coupling limit, we do not have a detector or amplifier but rather an attenuator,
which is of only academic interest to us.
Comparing the detector response (79) and back reaction part of Eq. (81), we see that
the mechanical oscillator behaves in the steady state as if in contact with a thermal
bath.8,9,12,26,37,38,39 The back reaction of the detector on the mechanical oscillator is effec-
tively that of a thermal bath with damping rate γback = γbm(Rγ − 1) and effective thermal
average occupation number nback defined as follows:
γback(2n
back + 1) = γbmN±. (83)
Thus,
n±back = (Rγ − 1)
. (84)
The failure to approach the lower quantum bound (75) when N± ≫ 1 then translates into
having (2n±back + 1)γback/γbm ≫ 1. Thus, to get close to the bound, we necessarily require
γback ≪ γbm;12 the back reaction occupation number n±back does not have to be small. With
the mechanical oscillator also in thermal contact with its external bath, the net damping
rate of the oscillator is γnet = γbm + γback = Rγγbm and the net, effective thermal average
occupation number nnet of the oscillator is defined as follows:
2n±net + 1
= γbm [2n(Rωωm) + 1] + γback
2n±back + 1
. (85)
Thus,
n±net = R
n(Rωωm) +
. (86)
From Eq. (78), we see that depending on the detuning parameter ∆ω = ωp − ωT , the
damping rate of the oscillator due to the detector back reaction can be either negative or
positive. Specifically, positive damping requires the following condition on the detuning
parameter:
∆ω < −
|c|2ω2TK2Tm
2I20ZpK
TmωTγpT
~ωm (γ
T +∆ω
. (87)
B. Displacement sensitivity
In the absence of a classical force acting on the mechanical oscillator, from Eq. (79) the
mechanical oscillator thermal noise displacement signal spectral density takes the familiar
Lorentzian form:
Sx(ω)|signal =
2Rγγbm
(ω − ωp ∓Rωωm)2 + (Rγγbm)2
2mRωωm
[2n (Rωωm) + 1] . (88)
In order to be able to resolve this mechanical signal, the detector noise (81) referred to the
mechanical oscillator input must be smaller than (88). The detector noise spectral density
at the input is
Sx (ω = ωp ±Rωωm)|noise =
Rγγbm
∓1 + |c|
2K2Tmω
γ2T + (∆ω ∓ ωm)
γ2T + (∆ω ± ωm)
|c|2K2Tmω2TγpT
2mRωωm
, (89)
where the first term on the right hand side is the back reaction noise acting on the mechanical
oscillator and the second term is the output, probe line zero-point noise referred to the input.
Note that the noise has been evaluated at ω = ωp±Rωωm, the maximum of the back reaction
Lorentzian.
If the detector output is to depend linearly on the mechanical oscillator signal input
(i.e., function as a linear amplifier), then back reaction effects must be small. In particular,
we require that γback ≪ γbm, i.e., Rγ ≈ 1. With |c| being proportional I0, we see from
Eq. (89) that increasing the drive current amplitude I0 increases the back reaction noise,
but decreases the probe line noise referred to the mechanical oscillator input. Thus, there is
an optimum I0 such that the sum Sx|noise is a minimum. Making the approximation Rγ = 1
and Rω = 1 in Eq. (89) and optimizing with respect to |c|, we find
Sx (ω = ωp ± Rωωm)|noise−optimum =
mωmγbm
∓1 + 2
γ2T + (∆ω ± ωm)
γ2T + (∆ω ∓ ωm)
. (90)
From Eq. (90), we see that the noise is further reduced if (i) the dominant source of trans-
mission line mode dissipation is due to energy loss through the coupled probe (information
gathering) line:12 γT ≈ γpT ; (ii) the detuning frequency is chosen to be ∆ω = ∓
γ2T + ω
where the minus (plus) sign corresponds to phase preserving (conjugating) detection. With
this detuning choice, the condition Rγ ≈ 1 requires (ωm/γT )2 ≪ 1 and so the minimum
detector noise is
Sx (ω = ωp ± Rωωm)|noise−optimum =
mωmγbm
2∓ 1 +O
(ωm/γT )
, (91)
where in order to determine the O ((ωm/γT )
2) term, the full form of Rγ given in Eq. (78)
must be used in Eq. (89) when optimizing. Comparing with Eq. (88) for the signal noise,
we see that to leading order the detector noise effectively doubles the zero-point signal in
1 1.5 2 2.5 3 3.5 4
I0 H10
-8 AL
FIG. 2: Displacement detector noise spectral density (solid line) and lower bound (dashed line)
versus drive current amplitude. The noise densities are evaluated at ω = ωp+Rωωm, corresponding
to phase preserving detection.
the phase preserving case. This exceeds the lower bound on the detector noise derived from
Eq. (75), which is zero to leading order in the phase preserving case.
We now numerically evaluate Eq. (89) for the detector noise. The feasible example
parameter values we use are:14 Bext = 0.005 Tesla, Zp = 50 Ohms, ωT/2π = 3×109 s−1, QT =
ωT/(2γT ) = 100, γT = 9.4×107 s−1, losc = 5 µm, λ = 1 (geometrical correction factor), m =
10−16 kg, ωm = 2.5× 107 s−1, and Qbm = ωm/(2γbm) = 103. These values give a mechanical
oscillator zero-point uncertainty ∆xzp = 1.45 × 10−13 m, a zero-point displacement noise
~/(mωmγbm) = 3.4 × 10−30 m2/Hz, and a dimensionless coupling strength KTm = −1.1 ×
10−5, where we assume that in the expression (22) for KTm, Φext can be chosen such that
the dimensionless factor Φ0
4πLT lIc
tan (πΦext/Φ0) sec (πΦext/Φ0) ≈ 1 (matching condition). We
also suppose that γT ≈ γpT , i.e., the transmission line mode damping is largely due to the
probe line coupling.
Fig. 2 shows Sx (ω = ωp +Rωωm)|noise × mωmγbm/~ and also the lower bound on the
detector noise that follows from Eq. (75) for phase preserving detection. Note that the
minimum detector noise is approximately 0.8 ~/(mωmγbm). Thus, for this example, the
next-to-leading O ((ωm/γT )
2) term in Eq. (91) is approximately −0.2. Note also that the
detector noise coincides with the lower bound in the small drive limit.
C. Force sensitivity
Consider a monochromatic classical driving force with frequency ω0 ∼ Rωωm acting on
the oscillator: Fext(ω) = F0δ(ω−ω0). The force signal spectral density is then SF (ω)|signal =
F 20 δ(ω − ω0). For force detection operation, the mechanical oscillator is included as part
of the detector degrees of freedom. From Eqs. (79-82), the force noise spectral density
evaluated at ω = ωp ± ω0 is
SF (ω = ωp ± ω0)|noise = 2m~ωmγbm
2n (ω0) + 1∓ 1 +
|c|2K2Tmω2TγT
γ2T + (∆ω ∓ ωm)
(ω0 −Rωωm)2 + (Rγγbm)2
γ2T + (∆ω ± ωm)
γbm|c|2K2Tmω2TγpT
. (92)
Comparing the displacement noise (89) with the force noise (92), we see that the latter
includes the additional 2m~ωmγbm [2n (ω0) + 1] mechanical quantum thermal displacement
noise term. Since the mechanical oscillator forms part of the force detector, it need not
necessarily be weakly driven and or weakly coupled to the transmission line; as explained in
Sec. IIIA, the present analysis employs a linear response approximation for force detection,
not displacement detection. Thus, in determining the optimum I0 (and or KTm) and ∆ω
such that SF |noise is a minimum, we should not assume a priori the restrictions Rγ, Rω ≈ 1.
2 4 6 8 10 12 14
I0 H10
-8 AL
FIG. 3: Force detector noise spectral density versus drive current amplitude for detuning ∆ω = 0
(solid line), ∆ω = −5ωm (dashed line), and ∆ω = −10ωm (dotted line) . The noise densities are
evaluated at ω = ωp +Rωωm, corresponding to phase preserving detection.
Fig. 3 shows the results of numerically evaluating the force noise spectral density given
by Eq. (92) for phase preserving detection (ω = ωp + ω0) and a range of detuning values.
The same example parameters are used as in the above displacement sensitivity analysis,
with n(ω0) = 0 and ω0 = Rωωm. The force noise is expressed in units 2m~ωmγbm = 6.6 ×
10−39 N2/Hz. Note that the minimum force noise is exactly 2 in these units, independently
of the detuning, with the minimum occuring at larger I0 values as the detuning is made
progressively more negative.
D. Back reaction cooling
From Eq. (86), we see that the net, thermal average occupation number nnet of the
mechanical oscillator’s fundamental mode decreases as Rγ increases. Thus, by increasing
the drive and or coupling strength such that γback ≫ γbm, the mechanical oscillator can be
effectively cooled at the expense of increasing its damping rate.3,8,9,21,26,40,41,42,43,44,45,46,47,48
Consider sufficiently negative detuning such that −∆ω ≫ |c|2ω2TK2Tm/(πωm) [see Eq. (87)].
Substituting definition (78) for Rγ and definition (82) for N+ into Eq. (86) and supposing Rγ
is large enough that we can neglect the external damping term γbm, we obtain approximately
for the phase preserving case:
n+net ≈
n (Rωωm)
+ n+back, (93)
where
n+back ≈ −
γ2T + (∆ω + ωm)
4∆ωωm
. (94)
This expression agrees with that derived in Ref. 26, apart from the 1/2 which is simply due
to a small difference in the way we define n±back in Eq. (83). Choosing optimum detuning
∆ω = −
γ2T + ω
m to minimize n
back in Eq. (94), we therefore have
n+net ≈
n (Rωωm)
1 + (γT/ωm)
2 − 1. (95)
How much cooling can be achieved depends on (i) how large Rγ can be, subject to the above
inequality on −∆ω; (ii) making the ratio γT/ωm as small as possible.26
Using the same example parameter values as above, but taking instead a larger but still
realistic quality factor Qbm = 10
4 for the mechanical oscillator,6 the resulting numerically
evaluated effective occupation number n+net [Eq. (86)] is given in Fig. 4 for a range of external
5 10 15 20 25 30 35 40
I0 H10
-8 AL
FIG. 4: Net effective average occupation number of the mechanical oscillator versus drive current.
The solid curve is for external bath temperature T = 100 mK [n(Rωω) = 523], the dashed curve is
for T = 10 mK [n(Rωω) = 52], and the dotted curve is for T = 1 mK [n(Rωω) = 4.8].
bath occupation numbers n(ωm). Thus, even for small coupling strengths KTm and drive
current amplitudes I0, significant cooling of the mechanical oscillator can be achieved. This
is in part a consequence of the fact that the quality factor Qbm of the mechanical oscillator
when decoupled from the detector is very large.
V. CONCLUDING REMARKS
In the present paper, we have attempted to give a reasonably comprehensive analysis
of the quantum-limited detection sensitivity of a DC SQUID for drive currents well below
the Josephson junction critical current Ic. In this regime, the SQUID functions effectively
as a mechanical position-dependent inductance element to a good approximation and the
resulting closed system Hamiltonian (24) takes the same form as that for several other
types of coupled mechanical resonator-detector resonator systems. Thus, the key derived
expressions (69) and (70) for the detector response and detector noise are of more general
application.
The main approximation made in analyzing the position and force detection sensitivity,
as well as back reaction cooling, was to limit the drive current and or coupling strength
according to Eq. (76). This allowed us to find much simpler, analytical approximations to
the key expressions, in particular Eqs. (79) and (81). The regime of larger drive currents and
or coupling strengths which exceed the limit (76) remains to be explored. However, with
the SQUID in mind, it is more appropriate to consider larger drive currents in the context
of including the non-linear I/Ic corrections to the SQUID effective inductance. This will be
the subject of a forthcoming paper.19
Acknowledgements
We thank A. Armour, A. Rimberg, and P. Nation for helpful discussions. This work was
partly supported by the US-Israel Binational Science Foundation (BSF) and by the National
Science Foundation (NSF) under NIRT grant CMS-0404031. One of the authors (M.P.B.)
thanks the Aspen Center for Physics for their hospitality and support where part of this
work was carried out.
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	Introduction
	Equations of Motion
	Transmission line-SQUID-mechanical oscillator Hamiltonian
	Open system Heisenberg equations of motion
	Observables and `in' states
	Solving the equations of motion
	Linear response approximation
	Semiclassical approximation
	Complete solution to detector signal response and noise
	Quantum bound on noise
	Results
	Analytical approximations
	Displacement sensitivity
	Force sensitivity
	Back reaction cooling
	Concluding remarks
	Acknowledgements
	References
