Download Quantum critical states and phase transitions in the presence of non

Quantum critical states and phase transitions in the presence of non equilibrium noise
Emanuele G. Dalla Torre1 , Eugene Demler2 , Thierry Giamarchi3 , Ehud Altman1
1
Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, 76100, Israel
2
Department of Physics, Harvard University, Cambridge MA 02138
3
DPMC-MaNEP, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva, Switzerland
(Dated: October 19, 2009)
Quantum critical points are characterized by scale invariant correlations and correspondingly long
ranged entanglement. As such, they present fascinating examples of quantum states of matter, the
study of which has been an important theme in modern physics. Nevertheless very little is known
about the fate of quantum criticality under non equilibrium conditions. In this paper we investigate
the effect of external noise sources on quantum critical points. It is natural to expect that noise
will have a similar effect to finite temperature, destroying the subtle correlations underlying the
quantum critical behavior. Surprisingly we find that the ubiquitous 1/f noise does preserve the
critical correlations. The emergent states show intriguing interplay of intrinsic quantum critical and
external noise driven fluctuations. We demonstrate this general phenomenon with specific examples
in solid state and ultracold atomic systems. Moreover our approach shows that genuine quantum
phase transitions can exist even under non equilibrium conditions.
I.
INTRODUCTION
An important motivation for investigating the behavior of non-equilibrium quantum states comes from state
of the art experiments in atomic physics. Of particular interest in this regard are systems of ultracold polar molecules[1, 2] and long chains of ultracold trapped
ions[3]. On the one hand these systems offer unique possibilities to realize strongly correlated many-body states,
which undergo interesting quantum phase transitions[4–
6]. But on the other hand they are controlled by large
external electric fields, which are inherently noisy and
easily drive the system out of equilibrium[7, 8]. It is natural to ask what remains of the quantum states, and in
particular, the critical behavior under such conditions.
The effect of non-equilibrium noise on quantum critical points is also relevant to more traditional solid-state
systems. Josephson junctions, for example, are known to
be affected by non-equilibrium circuit noise, such as 1/f
noise. Without this noise a single quantum Josephson
junction [9] should undergo a text-book quantum phase
transition[10, 11]: depending on the value of a shunt resistor, the junction can be in either a normal or a superconducting state. A phase transition occurs at a universal value of the shunt resistance Rs = RQ = h/(2e)2 ,
independent of the strength of the Josephson coupling.
This is closely related to the problem of macroscopic
quantum tunneling of a two level system (or q-bit) coupled to a dissipative environment (at equilibrium and
zero temperature) [12]. For weak dissipation, a particle placed in one of two wells will tunnel to the second
well. But beyond a critical value of the dissipation it undergoes a localization transition and will remain in the
first well eternally.
There is a large body of work on 1/f noise as a source of
decoherence for superconducting q-bits (see e.g. [13, 14]
and references therein). However the effect of such noise
on the quantum phase transitions and the non equilibrium steady states of Josephson junctions poses funda-
mental open questions. Do the different phases (superconducting or normal) retain their integrity in presence of
the noise? Is the phase transition between them sharply
defined?
In this regard it is worth noting that thermal noise
(that is T > 0) in equilibrium is always a relevant perturbation at a quantum critical point[15, 16]. In particular
there is no sharp distinction between the different phases
of the low dimensional systems we consider here at any
T > 0. In certain cases it was argued that a non equilibrium drive may act as an effective temperature[17, 18].
In marked contrast, we find that the external 1/f noise
is only a marginal perturbation at the critical point in
many cases of interest. It gives rise to a modified critical
point in which both quantum fluctuations and the external classical fluctuations play an important role. The
phases on the two sides of the transition remain well
defined, however their properties change in an interesting way because of the non equilibrium conditions. This
phenomenon is seen most clearly in the example of the
shunted Josephson junction subject to 1/f noise, which
we work out in Sec. II.
In section III we investigate the potentially richer
physics of one-dimensional systems. We focus on long
chains of ultracold polar molecules or of trapped ions.
Despite the extended interactions these systems cannot quite crystalize, even at equilibrium, because of the
strong quantum fluctuations in one dimension. At T = 0
they form critical liquids which display power-law crystalline correlations[19] (but see also footnote [20]). However, if the correlations decay sufficiently slowly, the liquid can be pinned by impurity potential of arbitrary
strength. Such pinning, either by a single impurity[21]
or a commensurate lattice potential[19] takes place, at
equilibrium, as a quantum phase transition that can be
tuned by the inter-particle interaction (For application to
ion traps see Ref. [6]). Another interesting phenomenon
in ion chains is the zigzag instability[22], which is expected to evolve into a true quantum phase transition in
2
the limit of long chains.
Again the relevant issue is the fate of these critical
states and quantum phase transitions in the presence of
noisy electrodes. Such noise has been characterized in recent experiments with ion traps[7, 8], where it was found
to have a 1/f power spectrum and attributed to localized charge patches on the electrodes. Using a Keldysh
approach we formulate an exact long-wavelength description of the steady state. A crucial result of this analysis
is that power-law correlations are maintained in presence
of the noise. The correlation exponent can be tuned continuously by the noise mimicking the effect of varying the
interaction (Luttinger) parameter.
The fact that the system is out of equilibrium is betrayed by its linear response to external probes. In particular we compute the response to a perturbation periodic in space and time (Bragg spectroscopy). Very much
as in equilibrium, we find an absorption spectrum (or
loss function) characterized by a power law, which reflects the correlations in the critical steady state. The
dramatic signature of the non-equilibrium conditions is
seen in the sign of the absorption function, which changes
from positive (loss) to negative (gain) as a function of the
noise power. is It is the constant pumping by the noise
field, that allows the probe field to extract energy from
the system in steady state.
The long wavelength description of the critical steady
state allows us to study its stability to various static perturbations within a renormalization group framework. In
this way we describe pinning by a static impurity and
by a lattice potential. We show that pinning-depinning
occurs as a phase transition driven by interplay of the
intrinsic quantum fluctuations and the external noise.
II.
In our discussion of the Josephson junction we consider
the standard circuit shown in Fig. 1(a). Note that the
charging energy of the junction (N − N0 )2 (e2 /2C) is determined by the junction capacitance and the deviation
from an offset charge. In realistic junctions, the offset
charge eN0 has random time dependent fluctuations with
a 1/f spectrum [23] hN0∗ (ω)N0 (ω)i = F0 /(|ω|/2π). This
is modeled by the fluctuating voltage source VN (t) =
eN0 (t)/C.
The classical equation of motion for the circuit in Fig.
1(a) is given by Kirchhoff’s current law:
~
C~
θ̇ +
θ̈ − C V̇N (t)
2eR
2e
1
1
cθ̈ + η θ̇ = ζ(t) + Ṅ0 (t).
2
2
(1)
The equation is easily derived from Josephson relations
IJ = J sin(θ) and VJ = (~/2e)θ̇. Note that the voltage
across the capacitor is VJ −VN , and so the two last terms
in Eq. (1) give the current on the capacitor. In what
follows we consider the case of zero bias (Iext = 0).
(2)
Here c = ~C/2e2 and η = (1/2π)RQ /R. Eq. (2) describes a damped harmonic oscillator. The random forcing term ζ(t) originates from the equilibrium bath. If the
resistor is maintained at zero temperature, this noise has
the power spectrum hζω? ζω i = η|ω|. The other random
forcing term is the time derivative of the charge noise.
Since the charge fluctuations have a spectrum ∼ F0 /|ω|,
the power spectrum of Ṅ0 is ∼ F |ω|, which mimics the
resistor noise. Unlike the resistor, however, external fluctuations do not have an associated dissipation term. This
is because the noise source is not in thermal contact with
the system. Thus the fluctuation dissipation theorem
is violated in the presence of the non-equilibrium noise
source. It would be respected of course if we had only
noise from the resistor.
For our subsequent analysis, and in particular to facilitate the Josephson coupling, it is easier to work within
the Keldysh framework (see e.g. [25]). In particular,
the linear equation (2) is equivalent to the following
quadratic quantum action within the Keldysh approach:
Z
−1
S0 =
dω θω† G0ω
θω
1
0
cω 2 − iηω
−1
2
G0ω =
π
1
2
2 cω + iηω −iη|ω| − i 4 F0 |ω|
∗
θω† ≡ (θcl,ω
θ̂ω∗ )
PHASE TRANSITION IN A NOISY
JOSEPHSON JUNCTION
Iext = J sin θ +
Weak coupling – To include quantum phase fluctuations, we first consider the system at vanishing Josephson coupling, which leaves us with a linear equation of
motion. Treating the resistor as an ohmic bath in thermal contact with the system[24] results in the quantum
Langevin equation:
(3)
Here θcl , and θ̂ are the ”classical” and ”quantum” fields.
As usual they are defined as the symmetric and antisymmetric combinations, respectively, of the fields associated with forward and backward time propagation of
operators: θcl = (θf + θb )/2, θ̂ = θf − θb . We note
that the capacitive term ∝ ω 2 is irrelevant, by power
counting, at low frequencies. By contrast, the contribution of the non equilibrium noise has the same scaling
dimension as the terms coming from the resistor. Therefore the fixed-point action consists of the contributions
of both the external noise and the resistor.
Using either the quadratic action (3) or the linear equation of motion (2), we can compute the phase autocorrelation function, which decays as a power-law at long
times
hcos [θcl (t) − θcl (0)]i ∼ t−(1+πF0 /4η)/πη
(4)
As expected, we see that the irrelevant charging term
does not influence the long time behavior of the correlations. The non equilibrium noise on the other hand
3
Iext
RQ/R
EJ/EC
1.5
1.5
!
superconductor
1
C
~ VN (t)
0.5
0.5
superconductor
0
0.5
insulator
1
F0
0
0.5
1
F0
insulator
insulator
superconductor
R
(d) Non equilibrium
phase diagram
R/RQ
1
J
(c) Strong coupling
(EJ >> EC )
(b) Weak coupling
(EJ << EC )
(a) Circuit
0
0
1
!
R/RQ
FIG. 1: (a)Electronic circuit relevant to a resistively shunted Josephson junction with charging noise. (b) Critical resistance
R/RQ as function of the noise strength F0 , in the weak coupling limit (Eq. (6)). (c) Critical conductance R/RQ as function of
the noise strength F0 , in the strong coupling limit. Figures (a) and (b) are related by the duality transformation R/RQ → RQ /R
and ”superconductor”↔”insulator”. (d) Schematic phase diagram at equilibrium (dotted line) and in the presence of nonequilibrium one-over-f noise (dashed line)
acts as a marginal perturbation. It does not destroy
the power-law scaling, but modifies the exponent. We
conclude that a critical (scale-invariant) non-equilibrium
steady state obtains in presence of the external noise.
The important question to address in the context of
weak coupling, is under what conditions the critical
steady-state we just described is stable to introduction of
the Josephson coupling as a perturbation to the action.
Since, in absence of the perturbation, we have a scale
invariant state, we can apply the usual RG program to
determine its fate in presence of the perturbation. That
is, we should find how the perturbation transforms under a scale transformation that leaves the critical steadystate invariant. In general, the action of the Josephson
coupling
Z
SJ = J dt [cos θf (t) − cos θb (t)]
(5)
is not scale invariant. From the decay of the correlation
function (4) we can directly read off the anomalous scaling dimension of the perturbation in the critical state,
which is α = 1 − (1 + 2πF0 /η)/2πη. When α > 0 the
perturbation grows under renormalization and ultimately
destabilizes the critical steady-state. We therefore predict a phase transition at a critical resistance
√
1
R∗
2π 2 F0 + 1 − 1
=
=
,
(6)
2πη ∗
RQ
π 2 F0
below which, the Josephson coupling term becomes relevant. Note that we recover the equilibrium dissipative
transition at R∗ = RQ in a ”quiet” circuit (F0 = 0).
We can tune across the transition also by maintaining
a constant resistance R < RQ and increasing the nonequilibrium noise ”power” F0 , as shown in Figure 1 (b).
Within the weak coupling theory we do not have direct
access to the properties of the steady-state at R < R∗ .
However because the Josephson coupling grows under
renormalization it is reasonable to expect that the junction would be superconducting. The phase would be essentially locked because of the large renormalized Josephson coupling effective at long time scales. In order to
determine with more confidence that the flow to strong
coupling indeed signals a superconducting phase we shall
now take the opposite, strong coupling viewpoint.
Strong coupling – In the strong coupling limit J >>
e2 /c we may employ a well known duality between weak
and strong coupling[10, 26]. Under the duality transformation the creation operator of a cooper pair on the
junction eiθ is mapped to the creation operator of a phase
slip eiφ . The quantum action for the phase slips contains two terms: a quadratic term of the same form as
(3), with θ → φ and 2πη → 1/2πη, and a cosine term
Sg = g cos(φ). The latter describes tunneling of phase
slips. The RG analysis can proceed in the same way
as above, giving a transition
at the value of shunt resis√
tance R∗ /RQ = π 2 F0 /( 2π 2 F0 + 1−1). For R < R∗ the
phase-slip action Sg is irrelevant. We therefore establish
the existence of a superconducting phase for R < R∗ , at
least in the strong coupling limit.
The combined results of the weak and strong coupling
analysis imply a phase diagram of the form shown in
Figure 1(d). At weak coupling the critical resistance,
in presence of noise, occurs at R∗ which is smaller than
RQ , while at strong coupling R∗ is larger than RQ . The
dashed line in this figure shows a simple interpolation of
the phase boundary between the two limiting regimes.
However, we cannot exclude the possibility that new
phases, such as a metallic phase arise at intermediate
coupling.
III.
ONE DIMENSIONAL CHAINS OF POLAR
MOLECULES OR TRAPPED IONS
We now turn to investigate the interplay between critical quantum fluctuations and external classical noise in
4
one-dimensional systems. Good laboratories for studying
such effects experimentally are provided by ions in Ring
or linear Paul traps, as well as Polar molecules confined to
one dimension. Because of the confinement to one dimension both systems are affected by quantum fluctuations.
On the other hand they are subject to noisy electric fields
that can influence the steady state correlations.
In ion traps, the fluctuations of the electric potential,
which couples to the ionic charge density, have been carefully characterized[7, 8]. The noise power spectrum was
found to be very close to 1/f and with spatial structure
indicating moderately short range correlations. In the
molecule system electric fields are used to polarize the
molecules, and fluctuations in these fields couple to the
molecule density via the molecular polarizability.
For the theoretical analysis we consider a one dimensional model of interacting bosons in the continuum, subject to external classical noise. Note however that in the
limit of strong extended interactions, when the particles
almost form a perfect crystal, the distinction between
bosons and Fermions is essentially non existent. Our
starting point is the universal low energy Hamiltonian describing long-wavelength density fluctuations (phonons)
in one dimensional systems[27]
Z
vs
1
H0 =
dx
(∂x φ)2 + πK(Π)2 .
(7)
2
πK
The hamiltonian is written in terms of the the continuum field φ(x, t), which represents the displacement of
the particles from a putative Wigner lattice they would
form if the interaction were infinitely strong. Π is the
canonical conjugate field and the constant vs is the sound
velocity which we shall set to 1 throughout. Finally K is
the Luttinger parameter determined by the microscopic
interactions.
Smooth (long wavelength) density fluctuation are represented by the gradient of the displacement field,
(−1/π)∂x φ(x, t). The part of the density with fourier
components of wavelengths near the inter-particle spacing are encoded by ÔDW = ρ0 cos(2φ(x, t))[27], where
ρ0 is the average density The former is therefore also
the density wave (or solid) order parameter field of the
S0 =
X (φ∗ φ̂∗ )
cl
ω,q
Wigner lattice.
In one dimension true solid order cannot exist. At
best, the equilibrium zero temperature correlations decay as a power law[19], hÔDW (x)ÔDW (0)i ∼ x−2K . The
lattice may become pinned in the presence of a perturbation, such as an impurity or a periodic potential. At
equilibrium this occurs as a quantum phase transition at
a critical value of the Luttinger parameter. As in the
Josephson junction we wish to address two questions; (i)
How does the external noise affect the the steady-state
power-law correlations; (ii) How does it influence phase
transitions, such as the lattice pinning transition.
We model the external electric noise as a random
time dependant field coupled to the particle density.
In general, the noise couples to both components of
the density via the terms −f (x, t)π −1 ∂x φ(x, t) and
ζ(x, t)ρ cos(2φ(x, t)). For now we assume that the noise
source is correlated over sufficiently long distances, that
its component at spatial frequencies near the particle
density (ζ(x, t)) is very small and can be neglected.
In this case the long wave-length theory remains harmonic. We shall characterize the noise by its power spectrum F (q, ω) = hf (q, ω)f (−q, −ω)i. We take this to be
1/f noise with short range spatial correlations, that is
F (q, ω) = F0 /(|ω|/2π).
When the system is irradiated with external noise we
expect it to absorb energy and heat up. In order to stabilize a steady state we need a dissipative bath that can
take this energy from the system. In the Josephson junction problem, the resistor naturally played this role. Is
there a similar dissipative coupling in the one dimensional
systems under consideration here?
In the ion traps, there is a natural dissipative coupling
because these systems can be continuously laser cooled.
Thus the system can reach a steady state, which reflects a
balance between the laser cooling and the external noise.
The polar molecules do not couple to a natural source of
dissipation, however a thermal bath can in principle be
realized by immersion in a large atomic condensate[28].
Altogether, the Keldysh action that describes the system with the external noise and the dissipative coupling
is given by
0
1
2
2
πK (ω − q ) + iηω
Here 2πF0 /|ω| is the external noise power. The factor
of q 2 /π 2 in front of this term appears because the noise
couples to (1/π)∂x φ in the hamiltonian. η denotes the
dissipative coupling to laser cooling. The cooling laser
applies a velocity dependent friction force on each ion
and therefore it couples to the ion displacement field as
1
2
πK (ω
− q 2 ) − iηω
q 2 F0
−iη|ω| − i 2π|ω|
!
φcl
φ̂
(8)
η φ̇.
Fundamentally, the above action is very similar to (3)
and consists of the natural extension from a single oscillator to a one dimensional chain. However there is
an important difference. The harmonic chain, is scale
invariant only without the noise and dissipation terms,
5
which are strictly speaking relevant perturbations of this
fixed point. Indeed, the dissipative coupling generates
a relaxation time-scale τ ∼ 1/η, which breaks the scale
invariance. To retain the scale invariance and still drive
the system out of equilibrium we can consider the interesting limiting regime in which both η → 0 and F0 → 0,
while the ratio F0 /η tends to a constant. Then the Crystal correlation function is easily calculated and seen to
be a power law
hcos(2φcl (x)) cos(2φcl (0))i ∼ x−2K(1+F0 /2πη) .
(9)
The same exponent holds for the temporal correlations.
We see that the dimensionless ratio F0 /η, which measures
the deviation from equilibrium, acts as a marginal perturbation. In practice η and F0 are non vanishing. Then the
result (9) will be valid at scales shorter than 1/η. Correlations will decay exponentially at longer scales. Thus
η serves as an infrared cutoff of the critical steady-state.
It becomes unimportant if other infrared cutoffs, such as
inverse system size (1/L) or the lower cutoff of the 1/f
noise spectrum are anyway larger than η, as may be the
case in practice.
The density-density correlations can be measured directly by light scattering. The (energy integrated) light
diffraction pattern in the far field limit gives directly the
static structure factor S(q) = hρ−q ρq i of the sample. In
particular, the power-law singularity in S(q) near wavevector q0 ∼ 2πρ0 is just the fourier transform of the power
law decay of the Wigner crystal correlations (9).
We can also compute the decay of phase correlations
by considering the dual representation of the harmonic
action (8). We find
hcos[θcl (x) − θcl (0)]i ∼ x
−(1+π 2 F0 /2πη)/2K
FIG. 2: Imaginary part of the density-density response function in a one dimensional system described by (7), with vs = 1
and K = 0.5: (a) At equilibrium, F0 = 0. (b) In the presence
of 1/f noise with F0 /2πη = 4.
exponent. If it is more singular, then the noise is relevant
and leads to correlations decaying faster than a powerlaw. Nevertheless if the noise power is close but not exactly 1/|ω| the critical power-laws (9) and (10) would still
be observed over a wide range of intermediate distances.
A.
Response
(10)
Again we see that the the noise power can be used to tune
the decay exponents. In a system of ultracold molecules
the phase correlations can be directly measured using
interference experiments [29].
It is interesting to compare the results (9) and (10) to
the equilibrium case (F0 = 0). In equilibrium the correlation exponent is controlled by the Luttinger parameter K
which is in turn determined by the interactions. Reducing K (by increasing interactions) leads to a slower decay of density-wave correlations and concomitantly faster
decay of phase correlations. This duality between phase
and density is a consequence of the fundamental uncertainty relation between them, which is saturated in the
harmonic ground state. The duality is destroyed under
the non equilibrium conditions. In particular, increasing
the noise leads to a faster decay of both the density and
phase correlations.
Before proceeding, a brief note on the case of external
noise with a spectrum different from 1/|ω|. The effect of
such noise on the critical state depends on its behavior
at low frequency. If it is less singular than 1/|ω|, then
the noise is an irrelevant perturbation and correlations
at sufficiently long distances decay with the equilibrium
Because of the non equilibrium conditions, the usual
fluctuation dissipation relation connecting correlations,
such as (9), and corresponding response functions, does
not hold. we shall see that this fact has dramatic implications on the nature of the system’s response to external
perturbation and in particular on the dissipation, or energy lost by the probe acting on the critical steady-state.
As a concrete example we shall consider the densitydensity response function, given within the Keldysh
framework by:
χ(x, t) = ihρ (φf (x, t)) [ρ (φf (x, t)) − ρ (φb (x, t))]i. (11)
The fourier-transform of this function in space and time
gives the linear response of the density of the system
to a dynamic perturbation consisting of a weak periodic
potential with wave-vector q, oscillating and a frequency
ω. This is exactly the response function probed by Bragg
spectroscopy[30–32].
In the continuum limit of the one dimensional systems we consider here, the response is separated into two
regimes. The response at small wave-vectors q << q0 =
2πρ0 involves the component of the density ρ0 (x, t) =
∂x φ effective at these wave-vectors. On the other hand,
6
the response at wave-vectors near the inverse interparticle distance involves the component of the density
ρq0 (x, t) = cos(2φ).
At small wave-vectors the response is independent of
the noise and unchanged compared to the equilibrium
case. In particular the imaginary part of the response
function at these wave-vectors is given by χ00 (q, ω) =
K|q|Θ(ω)δ(ω − q). This is hardly surprising. Since the
system is harmonic, any two perturbations that couple
linearly to the oscillator field simply add up independently by the superposition principle. Here we can think
of the random noise field and the oscillating potential as
two such perturbation.
The response at wave-vectors near q0 is much more
interesting because the perturbation ∝ cos(2φ) couples
non linearly to the oscillator field φ. Nevertheless we can
calculate this exactly using ρq0 (x) in Eq. (11) evaluated
in the action (8). The result is (see supplementary material)
χ00 (q, ω) = C (K, K? ) (ω 2 − δq 2 )K? −1 Θ(ω 2 − δq 2 )
1
sin(πK)
C(K, K? ) =
(12)
4Γ2 (K? ) sin(πK? )
Here δq ≡ q − q0 , and we have defined K? ≡ K(1 +
F0 /2πη). This response function is shown in Fig. 2,
which also includes the delta-function response found for
small wave vectors. By comparing the plot in panel (a),
showing the case of vanishing noise, to panel (b) where
F0 /2πη = 4, we see that the noise causes a significant
suppression of the low frequency response at q = q0 . In
this case the equilibrium response function was divergent
at low frequency (panel (a). However the noise was sufficiently strong to overcome this divergence and turn it
into power-law suppression at low frequencies (panel b).
In principle, one could achieve a similar suppression
of the response at low frequencies by weakening the repulsive interactions, that is, by increasing the Luttinger
parameter K. So, is the something in the spectrum (12)
that tells us that it comes from a system inherently out
of equilibrium?
To answer this, recall that the imaginary part of the
response function, is directly related to the dissipation
function Ė(q, ω) = ωχ00 (q, ω), or in other words, the rate
by which the probe is doing work on the system. Inspecting the pre-factor C(K, K? ) in Eq. (12) we find,
that depending on the noise power the work can be either positive or negative. This could never have been the
case in equilibrium! One cannot extract energy from a
system at equilibrium and therefore the energy dissipation function should be always positive.
The situation we encounter is analogous to the essential physics of a laser. The only way to achieve gain from
passing light through a medium is to pump the medium
out of equilibrium and achieve ”population inversion”. In
a laser the relevant population is that of two level system.
Therefore the gain spectrum is monochromatic, and appears at the frequency of the transition. Here by contrast
we have a many-body system in a critical steady state,
and the gain spectrum (12) reflects the critical exponents
of the steady state. The 1/f noise plays the role of the
pump in this case. To achieve gain the noise has to pass
a certain threshold. In fact we see in (12) a curious commensurability effect between the noise and the intrinsic
interactions that leads to oscillations between gain and
loss as a function of the noise power.
B.
Non equilibrium phase transitions
We have seen that by changing the 1/f noise one can
continuously tune the critical exponent associated with
the power-law decay of correlations in one dimensional
quantum systems. As in the case of the Josephson junction, we can ask if it is possible to use the new knob to
tune across a phase transition.
A text book[19] phase transition in one dimensional
quantum systems is that of pinning by a commensurate
periodic lattice potential. In equilibrium it occurs below a universal critical value of the Luttinger parameter
Kc = 2, regardless of the strength of the potential. In the
context of the real time dynamics, the periodic potential
is added as a perturbation to the action (8)
Z
Sp = g dxdτ [cos(2φf (x, t)) − cos(2φb (x, t))] (13)
The analysis now proceeds in exactly the same way as
for the Josephson junction at weak coupling. The scaling of the perturbation (13) in the critical steady state
is determined with the help of the correlation function
(9). We find that the action of the periodic lattice has
the scaling dimension αp = 2 − K(1 + F0 /2πη). This
implies an instability, which signals a phase transition to
a pinned state for (F0 /2πη) < 2K −1 − 1. In particular
for F0 = 0 we recover the equilibrium pinning (or Mott)
transition at the universal value of the Luttinger parameter Kc = 2. Note that for K > 2 the system is always
unpinned because F0 is non-negative.
A pinning transition can also occur in the presence of
a single impurity [21], represented by the action
Z
Si = Vi dτ [cos(2φf (0, t)) − cos(2φb (0, t))]
(14)
The main difference from the previous case is that this
perturbation is completely local and its scaling dimension
is reduced by 1 relative to the periodic potential: αi =
1 − K(1 + F0 /2πη). We see that the pinning is not as
strong as in the case of a periodic lattice. Accordingly
the de-pinning transition occurs at a lower critical noise
(F0 /2πη) = K −1 − 1, than in the case of the periodic
potential.
Note that at equilibrium the pinning by a single impurity is formally identical to the dissipative transition of
the Josephson junction. The harmonic Luttinger liquid
maps to a resistor with R/RQ = K, while the impurity
7
We described a new class of non-equilibrium quantum
critical states and phase transitions, which emerge in the
presence of external classical noise sources. Remarkably,
in certain important cases the non-equilibrium drive, acts
as a marginal perturbation at the quantum critical point.
Therefore, properties of the new critical states are determined by an interplay of both the intrinsic quantum
fluctuations and the external noise driven fluctuations.
This is in marked contrast to the effect of thermal noise
which invariably destroys quantum criticality.
The simplest example of the new critical behavior is
given by a resistively shunted quantum Josephson junction subject to 1/f charging noise. The 1/f noise is seen
to be a marginal perturbation at the transition between
a superconducting and an insulating junction. The critical resistance at which the transition occurs is modified
by the noise from the universal value R∗ = RQ found
at equilibrium to a non universal value, which depends
on the noise power and on the strength of the Josephson
coupling.
We proposed that the new critical points may also be
observed in one dimensional systems of ultracold trapped
ions or polar molecules. In a practical realization, such
systems would be driven out of equilibrium by 1/f electric field noise from the electrodes. We show that turning
on the noise does not destroy the power-law correlations,
but only modifies the decay exponent mimicking the effect of the interaction parameter.
There are, however, crucial differences from the equilibrium critical state. First, we find that the noise expedites the decay of both density and phase correlations,
thus destroying the well known duality between these
correlations, which exists at equilibrium. Second, the
dynamical response of the critical steady-state to an external probe field can betray a dramatic effect of the non
equilibrium conditions. Specifically the energy loss spectrum of the probe field becomes negative, turning from
loss to gain, for sufficiently strong drive by the noise field.
This is analogous to the gain achieved in a Laser when
the medium is pumped to achieve population inversion.
Here the gain spectrum has a critical power-law spectrum, rather than monochromatic as in a Laser.
We showed that quantum phase transitions, such as
pinning of the crystal by an impurity or by a commensurate lattice potential can take place in the presence
of the external 1/f noise. In particular the system can
be tuned across the depinning transition by tuning the
noise power. It would be very interesting to extend these
ideas to higher dimensional systems, such as one or two
dimensional arrays of coupled tubes of polar molecules.
The natural phases in equilibrium are the broken symmetry phases, either superfluid or charge density wave.
The intriguing sliding Luttinger liquid phase, which retains the one dimensional power-law correlations despite
the higher dimensional coupling, is expected to be stable
only in a narrow parameter regime[33]. Because the 1/f
noise acts to suppress both the phase and density correlations it will act to stabilize this phase in a much wider
regime.
Acknowledgements. We thank Erez Berg, Sebastian Huber, Steve Kivelson, Austen Lamacraft, Kathryn
Moler and Eli Zeldov for stimulating discussions. This
work was partially supported by the US-Israel BSF (EA
and ED), ISF (EA), SNF under MaNEP and division
II (TG), E. D. acknowledges support from NSF DMR0705472, CUA, DARPA, and MURI.
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