ARSAdvances in Radio ScienceARSAdv. Radio Sci.1684-9973Copernicus PublicationsGöttingen, Germany10.5194/ars-15-131-2017Markovian Dynamics of Josephson Parametric AmplificationKaiserWaldemarwaldemar.kaiser@tum.dehttps://orcid.org/0000-0001-9069-690XHaiderMichaelhttps://orcid.org/0000-0002-5164-432XRusserJohannes A.RusserPeterJirauschekChristianjirauschek@tum.deInstitute for Nanoelectronics, Technical University of Munich, Arcisstraße 21, 80333 Munich, GermanyWaldemar Kaiser (waldemar.kaiser@tum.de) and Christian Jirauschek (jirauschek@tum.de)21September20171513114023December201625August20176September2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://ars.copernicus.org/articles/15/131/2017/ars-15-131-2017.htmlThe full text article is available as a PDF file from https://ars.copernicus.org/articles/15/131/2017/ars-15-131-2017.pdf
In this work, we derive the dynamics of the lossy DC pumped non-degenerate
Josephson parametric amplifier (DCPJPA). The main element in a DCPJPA is the
superconducting Josephson junction. The DC bias generates the AC Josephson
current varying the nonlinear inductance of the junction. By this way the
Josephson junction acts as the pump oscillator as well as the time varying
reactance of the parametric amplifier. In quantum-limited amplification,
losses and noise have an increased impact on the characteristics of an
amplifier. We outline the classical model of the lossy DCPJPA and derive the
available noise power spectral densities. A classical treatment is not
capable of including properties like spontaneous emission which is mandatory
in case of amplification at the quantum limit. Thus, we derive a quantum
mechanical model of the lossy DCPJPA. Thermal losses are modeled by the
quantum Langevin approach, by coupling the quantized system to a photon heat
bath in thermodynamic equilibrium. The mode occupation in the bath follows
the Bose-Einstein statistics. Based on the second quantization formalism, we
derive the Heisenberg equations of motion of both resonator modes. We assume
the dynamics of the system to follow the Markovian approximation, i.e. the
system only depends on its actual state and is memory-free. We explicitly
compute the time evolution of the contributions to the signal mode energy and
give numeric examples based on different damping and coupling constants. Our
analytic results show, that this model is capable of including thermal noise
into the description of the DC pumped non-degenerate Josephson parametric
amplifier.
Introduction
Recent progress in fabrication of nanoelectronic devices and low-temperature
physics has increased the interest in superconducting quantum circuits. As
outlined in the Bardeen Cooper Schrieffer (BCS)
theory, superconductivity is based on the condensation of Cooper pairs.
introduced a macroscopic theory by describing
the superconducting phase by coherent matter waves, which exhibits
macroscopic quantum effects. Thus, fluctuations are very small, making
superconducting quantum circuits interesting for low noise devices. The basic
element in superconducting quantum circuits is the Josephson junction
predicted by . The Josephson effect predicts the
tunneling of Cooper pairs in two weakly coupled superconductors. Weak
coupling is achieved by a thin separating tunnel barrier
(). Josephson
junction based devices require a theoretical treatment using quantum
mechanics, as soon as the energy of a considered signal becomes as low as a
few microwave energy quanta, i.e. photons, at very low temperatures of only a
few Kelvin. Compared to a pure classical treatment, a quantum formalism also
considers effects due to spontaneous emission as well as induced quantum
noise by coupling the system to a Langevin heat bath.
Parametric amplification plays an important role in several physical
phenomena and is also utilized for amplification of electric signals. The
parametric amplifier amplifies the oscillating signal mode by coupling the
mode to an idler mode (;
; ). Strong coupling
is achieved by an oscillating non-linear coupling parameter. Radio-frequency
and microwave signals containing few quanta are weak compared to the noise
level of most detectors (). Recent experiments by
show, that parametric amplification based on the
Josephson junction faces quantum-limited amplification. So far, Josephson
parametric amplifiers (JPA) have been treated classically by
and . A quantum mechanical
model of the ideal JPA is derived by , which
is based on the model introduced by . Josephson
parametric amplifiers operating close to the quantum limit also require a
quantum mechanical treatment of losses.
In this work, we analyze the DC pumped non-degenerate Josephson parametric
amplifier (DCPJPA). Based on a classical description of the lossy DCPJPA, the
current noise correlation and the spectral power densities are derived. As
the classical model is not capable of describing all observable features of
the quantum-limited amplification regime, as e.g. spontaneous emission, there
is the need of a quantum mechanical description of the DCPJPA. Thus, we
derive a quantum mechanical model of the circuit including noise and losses.
Losses are considered using the quantum Langevin method, as outlined by
and . The resonator
circuits, i.e. the signal and the idler circuit, are coupled to a heat bath,
represented by a photon gas in thermal equilibrium. The heat bath induces
fluctuations in the resonator modes and causes damping of the signal energy.
The time evolution of the signal energy and the noise contributions are
derived based on the Heisenberg equations of motion. Simplification of the
Josephson coupling Hamiltonian is obtained by the rotating wave approximation
(RWA). Markovian dynamics neglecting memory-effects are assumed, which induce
white noise into the circuit (). We
explicitly compute the time evolution of the energy contributions for
different initial setting. Our analytic results show, that the considered
quantum mechanical model is capable of including thermal noise into the
description of the Josephson parametric amplifier. For low damping, the
signal energy is also amplified exponentially as shown in previous
publications by , damping reduces the
amplification of the energy contributions. Furthermore, we derive the quantum
mechanical current noise correlation resulting from the coupling to the heat
bath and link it to the classical conductance.
The quantum mechanical treatment of a lossy DCPJPA can be very useful when it
comes to detecting e.g. single microwave photons, where one is interested in
the quantum noise added by the amplification process. Standard solid-state
based amplifiers cannot be used at such low energies, because of their
relatively high thermal noise, compared to superconducting parametric
amplifiers. This could enable interesting applications in superconducting
quantum computing, where single radio-frequency or microwave photons interact
with qubits, themselves consisting of Josephson elements. The theoretical
treatment of noise brought in by the environment is crucial for the
realization and operation of superconducting quantum circuit based systems.
However, there are still a lot of things left open in the following
discussions. First of all, the Josephson junction is considered ideal in the
following, i.e. it is modeled as an ideal tunnel junction, without any
effective dissipation mechanisms .
Furthermore, a Markovian assumption is made on the heat bath coupling
mechanism in the signal and idler modes of the considered amplifier system,
i.e. the coupling of the system and the heat bath only depends on the current
state.
The Josephson Effect
provided the theory, that superconductivity
originates from the pairing of electrons with opposite spin and wave vector
to Cooper pairs. The superconducting ground state can be described by a
macroscopic matter wave function given by . The Josephson
effect is observed in two weakly coupled superconductors, seperated by a thin
tunnel barrier ().
The superconducting tunneling current iJ(t) is described by the
first Josephson equation given by
iJ(t)=Icsinφ(t),
where Ic is the critical Josephson current and φ(t) is
the quantum phase difference between both superconductors. The second
Josephson equation relates the quantum phase difference φ to the
applied voltage over the Josephson junction v(t) by
∂φ∂t=2πΦ0v(t),
with the magnetic flux quantum
Φ0=h2e≈2.0678×10-15Vs,
with the elementary charge e and the
Planck's constant h. A DC voltage V0 applied to the junction gives
rise to an AC current oscillating with the Josephson frequency
f0=2eV0h=483.6⋅V0GHzmV,
with the elementary charge e and the Planck's constant h. Introducing the
magnetic flux Φ(t) as the integral of the voltage v(t) over time, the
energy wJ(t) stored in a Josephson junction is given by
wJ(Φ(t))=WJ1-cos2πΦ(t)Φ0,
with the maximum Josephson energy WJ=Φ0IJ/2π.
Thus, the ideal Josephson junction is non-dissipative. The energy can be
considered as stored in the non-linear, lossless time-variable Josephson
inductance LS(t) defined by
LS(t)=Φ02πIccos(φ(t)).
Different from other inductors, it is possible to apply a DC voltage to the
Josephson junction. Although the flux Φ(t) and with it the quantum phase
difference φ(t) are going to infinity with time when a DC voltage is
applied, according to Eq. () the energy
wJ(t) stored in a Josephson junction remains bounded. When a DC
voltage is applied to the Josephson junction LS-1(t) varies
sinusoidally with time. Hence, the Josephson junction can be used as the
required non-linear time-variable susceptance for parametric amplification as
shown by and .
Classical Model of the Lossy Josephson Parametric Amplifier
Equivalent circuit of the lossy DCPJPA.
In this section, we outline the classical model of the lossy DCPJPA described
by . The negative resistance three-frequency
DCPJPA consists of a DC pumped Josephson junction oscillating with a
frequency f0 according to Eq. (), coupled to two
resonator modes with resonant frequencies f10 and f20 such that
f10+f20=f0 is fulfilled. Coupling a signal with frequency f1≈f10 into the signal circuit by mixing in the Josephson junction
in the idler circuit a signal at frequency f2 is excited fulfilling f1+f2=f0.
The equivalent circuit of the DCPJPA is shown in Fig. .
The inductor L1 and the capacitor C1 constitute the signal
resonator with resonant frequency f10, L2 and C2 represent the
idler resonator mode with resonant frequency f20. Classically, losses
are modeled by conductances G1 and G2. The thermal noise generated
by the conductances G1 and G2 is represented by the equivalent
noise current sources in1(t) and in2(t) and is1(t) is the
signal current source.
The relation between signal, idler and pump frequency is crucial for the
device characteristics. The general energy relations of the circuit shown in
Fig. are derived by . If the junction
is biased by a DC voltage V0 such that f0=f10+f20, the signal,
idler and pump energies obey the Manley-Rowe relation, derived by
, including an additional DC term according to
P1f1=P2f2=-P0f1+f2.
This energy relation indicates the possibility of non-degenerate DC-pumped
parametric amplification. Other choices of the relation between the different
mode frequencies and the pump frequency can result in other operational modes
such as a degenerate parametric amplifier or an up-converter
(). Since the power flow at f1 and f2
has opposite direction of the DC power flow at f0, the Josephson junction
impedance at f1 and f2 exhibits a negative real part, i.e. the
magnitude of the reflection factor is greater than 1. In this case the DCPJPA
acts as a negative-resistance reflection amplifier at f1. When the signal
is coupled out at f2 an additional power gain f2/f1 is obtained.
For the lossy DCPJPA the classical equations of motion are given by
dQ1(t)dt=Φ1(t)L1-G1Q1(t)C1+is1(t)+in1(t)+Icsinω0t+2eIcℏΦ1(t)+Φ2(t)cosω0t,dQ2(t)dt=Φ2(t)L2-G2Q2(t)C2+in2(t)+Icsinω0t+2eIcℏΦ1(t)+Φ2(t)cosω0t,dΦ1(t)dt=-Q1(t)C1,dΦ2(t)dt=-Q2(t)C2.
The angular frequency ω0=2πf0 is determined via
Eq. () by the applied DC voltage V0.
The conductances G1 and G2 yield a decay of the signals in the
signal- and idler circuits. The thermal excitation does not decay, since it
is regenerated due to the dissipation-fluctuation
theorem p. 151 by thermal fluctuations in
the conductances. Considering the thermal noise sources in1(t) and
in2(t) as classical Langevin noise sources,
see p. 143, they exhibit the correlation
functions c11(τ) and c22(τ), respectively
c11(τ)≡in1(t)in1(t′)=4kBTG1δ(t-t′),c22(τ)≡in2(t)in2(t′)=4kBTG2δ(t-t′),
where … denotes the ensemble average, kB
is the Boltzmann constant and T is the absolute temperature of the lossy
resonator.
The parametric amplification of the DCPJPA according to
Fig. is treated in detail in .
There the maximum available power gains gp11max and
gp22max of the negative resistance parametric amplifier operated
at frequencies f1 and f2, respectively, the maximum available power
gains gp21max of the up-converter and gp12max of the
down-converter are obtained following as
gp11max=gp22max=4(1-η)2,gp12max=f1f24η(1-η)2,gp21max=f2f14η(1-η)2,
with
η=e2Ic2h2f1f2G1G2.
To achieve high gain η has to be close to 1. If η reaches 1, the
parametric amplifier becomes unstable. We assumed single conductances G1
and G2 in the signal and idler circuits, respectively. Splitting the G1
into a generator conductance, a loss conductance and a load conductance
reduces the gain, see and
.
The Fourier transform of Eqs. () and () yields the
autocorrelation spectra of the noise current sources
C11(ω)=2kBTπG1,C22(ω)=2kBTπG2.
The available noise power spectral densities of G1 and G2, respectively
are
CPna=12πkBT.
with Eqs. () to () this yields the spectral power
densities CPn1 and CPn2 in the signal and idler
circuits,
CPn1(ω)=2π1(1-η)2+f1f21η(1-η)2kBT,CPn2(ω)=2π1(1-η)2+f2f11η(1-η)2kBT.
In the classical limit the idler noise contribution to the noise in the
signal circuit is down-converted by a factor f1/f2.
We obtain the semiclassical approach for the quantum noise when replacing
kBT→hf1-exphfkT.
Inserting this into Eqs. () and () yields
CPn1(ω)=2hf1π(1-η)211-exphf1kT+η1-exphf2kT,CPn2(ω)=2hf2π(1-η)211-exphf2kT+η1-exphf1kT.
In the limit hf≪kT these equations yield Eqs. () and
() whereas in the limit hf≫kT we obtain
CPn1(ω)=21+ηπ(1-η)2hf1,CPn2(ω)=21+ηπ(1-η)2hf2.
Quantum Mechanical Model of the Lossy Josephson Parametric Amplifier
In this section, we outline the quantum mechanical model of the lossy DCPJPA
described by and
.
Here, we have used a second order Taylor expansion for the energy stored in
the nonlinear Josephson inductance. We expressed the equations of motion in
terms of signal and idler charges and fluxes, since they form a set of
conjugate variables which will be useful later, when stating the quantum
mechanical Hamiltonian.
In a quantum mechanical model dissipation and fluctuations which are modeled
in the classical model by the loss conductances G1 and G2 and the
related noise current sources in1 and in2 are modeled by coupling
the conservative quantum system formed by the capacitors, inductors and the
Josephson junction to a so-called heat bath. This heat bath is a
many-electron system of electrons in thermal equilibrium. The losses
originate from the transfer of energy to this many-electron system where the
transferred energy is randomized. Vice-versa fluctuations from the heat bath
are coupled into the resonant circuits as outlined by
and .
The total Hamiltonian of the lossy DCPJPA is given by
H=HLC+HJJ+HB+HC,
where the unperturbed Hamiltonian HLC represents the
ideal lossless LC resonant circuits, the operator HJJ describes
the energy of the DC biased Josephson junction, HB
represents the heat bath describing a photon field in thermodynamic
equilibrium and HC gives the coupling between the LC
resonators and the heat bath. We apply the second quantization formalism
outlined by and . We
introduced the annihilation operator ai and the creation
operator ai† given by
ai=12ℏωiLiΦi+iωiLi2ℏQi,ai†=12ℏωiLiΦi-iωiLi2ℏQi,
where ℏ=h/(2π) is the reduced Planck constant, and
Φi and Qi are the flux and charge
operators, respectively, of the ith LC circuit. The flux and charge
operators fulfill the canonical commutation relation
Φi,Qj=iℏδij,
which gives the bosonic commutation relation
ai,aj†=δij.
With Eqs. () and () the system
Hamiltionans are given by
HLC=ℏω1a1†a1+12+ℏω2a2†a2+12,HJJ=WJ1-cos[ω0t+λ1(a1+a1†)+λ2(a2+a2†)],
where the dimensionless parameter λi of the ith LC circuit
coupled by the Josephson junction is defined by
λi=2απZF0Zi.
Here, α is the Sommerfeld's constant given by
α=e22cε0h≈1137.036.ϵ0 is the vacuum permittivity, c is the speed of light,
Zi=Ci/Li is the characteristic impedance of the i-th LC
circuit, and ZF0 is the free-space wave impedance.
The rotating wave approximation (RWA) is commonly used in analytical
treatments of quantum optical systems. According to ,
the system perspective is changed into a rotating frame. Rapidly rotating
terms are neglected, which delivers a valid approximation in case of near
resonance terms and low intensities (). Applying the
RWA for the DCPJPA yields
HJJNR≈γja1†a2†exp(-iω0t)+a1a2exp(iω0t),
where γj=WJλ1λ2 is the Josephson
coupling parameter describing the coupling strength between both resonators.
The energy of the ideal DCPJPA rises exponentially with time, as derived by
, and , describing a
parametric oscillation rather than a parametric amplification. Applying the
Langevin method we can treat losses of quantum mechanical systems by coupling
the system to a heat bath as proposed by and
. A heat bath is described by a large system of
closely spaced harmonic oscillators in thermodynamic equilibrium. In Fig.
the heat bath modes are represented by dashed modes
inside the frame. The Hamilton operators of the heat bath and the coupling
between the system and the bath are given by
HB=∑kℏω̃kbk†bk+12,HC=∑i=12∑k12ℏκikaibk†+bkai†,
where bk is the bath operator with angular resonance frequency
ω̃k. The coefficients κik are of microscopic
origin, describing the coupling strength between the ith resonator and the
heat bath mode k. The Hamilton operator HC describes
the so-called heat-bath, a large phonon system in thermal equilibrium. Energy
from our conservative system under consideration (SUC) is coupled into the
heat bath and dissipated by distribution over the phonon system. By this way
the energy is randomized. Vice-versa, since the coupling between the SUC and
the heat bath is reciprocal and obeys detailed balance, fluctuations are
coupled from the heat bath into the SUC. This is stated by the
fluctuation–dissipation theorem which applies both to classical and quantum
mechanical systems. Whereas in the classical consideration in the previous
chapter dissipation and fluctuations simply could be modeled
phenomenologically applying a conductance and an equivalent noise source, the
quantum mechanical treatment requires a micro-physical model using the
quantum mechanical many-body system of the heat bath. In Sect. 5, the
relation between the coupling coefficients κik and the classical
circuit parameters is derived.
Quantum Langevin Equations
The time evolution of the system operators is computed in the Heisenberg
representation using the Heisenberg equations of motion
ddtOH=iℏHH,OH,
where OH is any arbitrary operator describing the
system. Using the Hamiltonian defined in Eq. (), the
Heisenberg equations of motion are given by
da1Hdt=-iω1a1H-iγjℏa2H†e-iω0t-i∑kκ1k2bkH,da2Hdt=-iω2a2H-iγjℏa1H†e-iω0t-i∑kκ2k2bkH,dbkHdt=-iω̃kbkH-i∑i=12κik2aiH.
Here, the bosonic commutator relations described by
and are applied. As we
are only interested in the dynamics of the circuit operators, we eliminate
the bath degrees of freedom by integrating Eq. ().
Hereby, we obtain
bkH(t)=bkH(0)e-iω̃k(t-t0)-i∑i=12κik2∫t0taiH(t′)e-iω̃k(t-t′)dt′.
Replacing the integration variable t-t′=τ yields
bkH(t)=bkH(0)e-iω̃k(t-t0)-i∑i=12κik2∫0t-t0aiH(t-τ)e-iω̃kτdτ.
The initial time t0 is set to zero without loss of generality.
Substituting Eq. () into Eq. () gives
da1H(t)dt=-iω1a1H(t)-iγjℏa2H†(t)e-iω0t+f1(t)-∑k∑i=12κ1k2κik2∫0taiH(t-τ)e-iω̃kτdτ.
In this equation we have introduced the Langevin noise operator in terms of
the initial bath operators
fi(t)=-i∑kκik2bkH(0)e-iω̃kt.
The operator fi(t) represents a stochastic term with white
noise and Gaussian properties as described in detail by
. In the following, we denote
bkH(0)=bkH. We further simplify
the equations of motion by changing to the slowly varying operators defined
by AiH(t)=eiωitaiH(t).
Equation () is given in the slowly varying frame by
dA1H(t)dt=-iγjℏA2H†(t)+f1(t)eiω1t-∑k∑i=12κ1k2κik2×∫0tAiH(t-τ)e-i(ω̃k-ωi)τei(ω1-ωi)tdτ.
This equation can be simplified by applying the Markovian approximation. As
outlined by , in the Markovian approximation the
system depends only on its current state, thus we obtain
dA1H(t)dt≈-iγjℏA2H†(t)+f1(t)eiω1t-∑k∑i=12κ1k2κik2×AiH(t)∫0te-i(ω̃k-ωi)τei(ω1-ωi)tdτ.
We use according to ∫0tdτexp[-i(Ω-ω)τ]=πδ(Ω-ω)-PiΩ-ω,
with the Cauchy principal part P. Assuming a dense distribution
of heat bath modes in the frequency spectrum, we can replace the summation
over the modes by an integral over the density of states
∑k→∫0∞dωD(ω).
In case of the DCPJPA, we assume a photon heat bath in thermal equilibrium
described by the density of states
D(ω)=Vω2π2c3,
with the volume of the heat bath V. Applying this relation to
Eq. () we obtain the
rate equations
dA1H(t)dt=-12γ̃11A1H(t)-iγjℏA2H†(t)+g1(t)-12γ̃21A2H(t)eiω1-ω2t,dA2H(t)dt=-12γ̃22A2H(t)-iγjℏA1H†(t)+g2(t)-12γ̃12A1H(t)eiω2-ω1t,
with the noise operator gi in the slowly varying frame defined
by
gi=fieiωt.
Hereby, we introduced the damping matrix γ̃ij given by
γ̃=π2κ12ω1D(ω1)κ1ω1κ2ω1D(ω1)κ1ω2κ2ω2D(ω2)κ22ω2D(ω2).
If the noise operators gi are uncorrelated, we can drop the
off-diagonal damping terms γ̃ij. Experimentally, the
diagonal damping terms γ̃ii are observable by measuring the
linewidth of the resonance curve of the ith LC resonator. Following
, an input signal
bSin(t) can be introduced in the rate
equations with the coupling factor
γS=1RSC,
with RS as the impedance of the transmission line, from which the
signal is coupled to the LC resonant mode.
We obtain the rate equations of the photon annihilation and creation
operators
dA1H(t)dt=-12γ̃11A1H(t)-iγjℏA2H†(t)+g1(t)+γSbSin(t),dA1H†(t)dt=-12γ̃11*A1H†(t)+iγj*ℏA2H(t)+g1†(t)+γSbSin†(t),dA2H(t)dt=-12γ̃22A2H(t)-iγjℏA1H†(t)+g2(t),dA2H†(t)dt=-12γ̃22*A2H†(t)+iγj*ℏA1H(t)+g2†(t).
Following , the output field operator can be
obtained by
bSout(t)=γSA1H(t)-bSin(t).
In this work, we consider the input signal as an initial occupation of the
signal mode and derive the dynamics of the internal amplifier mode
A1H. According to Eq. (), the output signal
operator bSout only depends on the signal mode
operator A1H for a known input
bSin. Thus, an increase of the expectation
value in the signal operator A1H or equivalently in the
signal mode energy is required in order to amplify the input signal.
Solving Eqs. ()–(), not considering the input signal
operator bSin yields
A1H(t)=ξ11(t)A1H(0)+ξ12(t)A2H†(0)+∑kκ1kbkH-α+(ω̃k)+α-(ω̃k)ei(ω1-ω̃k)t+∑kκ1kbkHα+(ω̃k)e-γt4-α-(ω̃k)eγt4e-t4(γ̃11+γ̃22*)+∑kκ2k*bkH†-β+(ω̃k)+β-(ω̃k)e-i(ω2-ω̃k)t+∑kκ2k*bkH†β+(ω̃k)e-γt4-β-(ω̃k)eγt4e-t4(γ̃11+γ̃22*),
with
ξ11(t)=coshtγ4+γ̃22*-γ̃11ηsinhtγ4×e-t4γ̃11+γ̃22*,ξ12(t)=-4γjiγℏsinhtγ4e-t4γ̃11+γ̃22*,α±(ω)=iγ±γ-γ̃22*+γ̃11γ̃11+γ̃22*±γ+4i(ω1-ω),β±(ω)=4γjℏγ1γ̃11+γ̃22*±γ-4i(ω2-ω).
Here, we have introduced the effective driving parameter
γ=(γ̃11-γ̃22*)2+16|γj|2ℏ2.
We can directly link the energy expectation value in the signal circuit to
the expectation value of the number operator nLC,1 by
ELC,1=ℏω1nLC,1+1.
In analogy to the computed expectation value of the
number operator is described by
nLC,1(t)=A1H†(t)A1H(t)=nSignal(t)+nIdler(t)+nNoise(t).
The signal photon number is given by
nSignal(t)=ξ11*(t)ξ11(t)nSignal(0).
The photon number of the down-converted idler noise is described by
nIdler(t)=ξ12*(t)ξ12(t)nIdler(0)+1,
and the noise photon number rising from the coupling of the signal mode to
the heat bath
nNoise(t)=2γ̃11n‾(ω1)α(ω1)2+2γ̃22n‾(ω2)+1β(ω2)2+γ̃22n‾(ω2)+14πe-γt4β+(ω2)-eγt4β-(ω2)2β(ω2)e-γt4β+(ω2)-eγt4β-(ω2)+e-γt4β+(ω2)-eγt4β-(ω2)2e-t4(γ̃11+γ̃22*)+γ̃11n‾(ω1)4πe-γt4Rα*(ω1)e-γt4α+(ω1)-eγt4α-(ω1)+e-γt4α+(ω1)-eγt4α-(ω1)2e-t4(γ̃11+γ̃22*).
Hereby, we defined
α(ω̃)=α-(ω̃)-α+(ω̃),β(ω̃)=β-(ω̃)-β+(ω̃).
Even if both the signal and the idler mode are unexcited initially, i.e.
nSignal(0)=nIdler(0)=0,
the signal mode receives quanta by means of spontaneous emission resulting by
Eq. (). Both the down-converted idler noise
(Eq. ) and the noise photons (Eq. )
represent noise contributions to the signal mode.
In order to obtain the current noise correlation, we need to derive
c11(τ)=in1(t)in1(t′).
The coupling of the LC circuit to the heat bath is a phenomenological method
to induce noise and damping into the system, such that the noise currents are
not obvious from Eqs. ()–(). Therefore, we calculate
the correlation of
c11(τ)=ddtQ1(t)ddtQ1(t′),
and only keep the components representing the induced noise. Following
, we use the expectation values
AiH(t)gi†(t′)=γ̃ii2n‾(ωi),AiH†(t)gi(t′)=γ̃ii2n‾(ωi),gi†(t)gi(t′)=γ̃iin‾(ωi)δ(t-t′),gi(t)gi†(t′)=γ̃iin‾(ωi)+1δ(t-t′),g1(t)A2†(t′)=g1†(t)A2(t′)=0.
The occupation number of the heat bath at initial time is given by the
Bose-Einstein statistics
n‾(ωk)=bkH†bkH=eℏωk/(kBT)-1-1.
The resulting noise current correlation is given by
c11(τ)=ℏω1L1γ̃11n‾(ω1)+12δ(t-t′).
One can see, that the coupling of the heat bath to the resonant circuits
models noise current resulting from Langevin noise sources. The square-root
of the damping constant γ1 gives the coupling of the Langevin noise
source to the circuit, while n‾(ω1) is the occupation of
the heat bath at resonance frequency ω1. In contrast to the classical
Langevin noise source, even if no quanta occupy the heat bath mode with
resonance frequency ω1, noise is induced into the circuit. This is an
explicit consequence of the quantum mechanical treatment including the vacuum
field, approving the need of a quantum mechanical description of the DCPJPA.
Comparison of the classical with the quantum mechanical correlation function
yields the relation
4G1↔Vκ12(ω1)L1π2c3.
The quantum mechanical coupling constant determines the classical, observable
conductance.
Numerical Evaluation of the Dynamic Behavior
We give numerical examples on the dynamics of the expectation value of the
signal energy of the DCPJPA described in
Eqs. ()–(). For comparison with
, we set the frequency of the signal circuit to
f1=2.5GHz and the idler frequency f2=ω2/(2π)=7GHz, and T is chosen as the liquid helium temperature
T=4.2K.
Figure shows the time evolution of all contribution to the
normalized signal mode energy for a high damping. Hereby, we chose the
damping constants
γ̃11=γ̃22=2π×108s-1
and a Josephson current of IJ=0.5µA. The
signal energy decays directly, while the down-converted idler noise rises at
the beginning of the consideration. After 3ns the idler noise
energy decays with the same rate as the signal energy. The noise contribution
resulting from the coupling of the heat bath to the signal mode rises all the
time, but increases significantly after 40ns. The lower bound of
the total energy is given by 1/2ℏω1, due to the vacuum
fluctuations in the signal mode.
Time evolution of the signal mode energies for strong damping.
Time evolution of the signal mode energies for strong driving.
In Fig. the energy terms are represented for a strong
driving. Strong driving is achieved, if γ exceeds the sum of the
damping constants γ̃11+γ̃22. Here, the
pumping by the Josephson junction, indicated by γj, is strong enough
to overcome the damping. The regime can be controlled by the Josephson
current Ic. Compared to the settings in Fig. ,
we chose the Josephson current as Ic=2µA.
All the energy terms rise exponentially. The signal energy represents the
dominating energy contribution, while the down-converted idler noise and the
noise induced by the heat bath follow the signal energy with a delay in time.
The energy terms are unbounded, thus the quantum Langevin equations are not
capable of modeling any saturation effects in the signal mode. For parametric
amplification it is desirable to obtain a steady-state solution. The derived
formalism does not yield saturation in the signal mode energy, but rather
shows parametric oscillation. Considering Eq. (), the expectation
value of the output field operator increases with rising internal signal mode
A1H. Driving needs to overcome the dissipation in order
to amplify a given signal.
In Fig. , we show the temperature dependency of the noise
energy as a function of time. A Josephson current of Ic=1µA is assumed. Figure clearly
fulfills the expectation, that the noise contribution to the signal mode
rises with temperature. Higher temperatures are neglected, as many
superconducting materials only contain their superconducting phase below
their critical temperature, which for many materials is in the range of
liquid helium temperature.
Compared to the dynamics outlined in , even for
the strong driving configuration, the signal energy and the down-converted
idler noise are strongly damped. Furthermore, the model derived in
did not show a decay of the signal energy for
strong a damping.
Temperature dependency of the noise energy.
Conclusions
In this paper, we have investigated the Markovian dynamics of the DC pumped
non-degenerate Josephson parametric amplifier. We modeled losses in the
DCPJPA using the quantum Langevin approach. Hereby, the resonators, i.e. the
signal circuit and the idler circuit, are coupled to a photon heat bath in
thermodynamic equilibrium. The DC pump voltage induces an oscillating
Josephson current, which is required for coupling of the resonators. We
outlined the classical and quantum mechanical model of the lossy DCPJPA and
derived the Heisenberg equations of motion. Dissipated signal energy is
randomized in the heat bath, for large damping the heat bath injects large
noise into the circuit. The expectation value of the signal and noise
energies is derived and numerically evaluated for specific settings. Strong
damping showed a significant decay of the signal energy and the rise of the
noise resulting by the coupling to the heat bath. Moreover, the temperature
dependency of the noise energy has been evaluated. The quantum Langevin
approach induces dissipation and noise into the dynamical behavior of the
DCPJPA, but does not cause saturation in the signal energy.
give a phenomenological model, in which damping is
induced by a multi-photon coupling. Hereby, saturation is reached, but still
a theory directly including saturation is desirable. Since many
approximations have been used in order to derive an analytic solution, the
origin of the missing saturation might be the Markovian assumption itself or
the leading order approximation of the cosine in the Josephson Hamiltonian.
No data sets were used in this article.
The authors declare that they have no conflict of
interest.
This article is part of the special issue “Kleinheubacher
Berichte 2016”. It is a result of the Kleinheubacher Tagung 2016,
Miltenberg, Germany, 26–28 September 2016.
Acknowledgements
Christian Jirauschek acknowledges funding by the Heisenberg program of the
German Research Foundation (DFG, JI115/4-1).
This work was supported by the German Research Foundation (DFG) and the Technische Universität München within the funding programme Open Access Publishing.
Edited by: Dirk Killat Reviewed by: two anonymous referees
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