+\Gamma(\omega-I\Phi_{0})+\Gamma(-\omega+I\Phi_{0})+\Gamma(-\omega-I\Phi_{0})\bigr{)},$
(202)

where we introduced the function

$\Gamma^{R}(\omega)=\frac{\gamma_{QPS}^{2}}{4}\int\limits_{-L/2}^{L/2}dxdx^{\prime}\bigl{(}P(x,x^{\prime};\omega)-P^{*}(x^{\prime},x;-\omega)\bigr{)}$
(203)

related to $\Gamma(\omega)$ (193) as

$\Gamma^{R}(\omega)=\int\frac{dz}{2\pi
i}\frac{\Gamma(z)-\Gamma(-z)}{z-\omega-i0}.$ (204)

For an illustration let us consider a short superconducting nanowire embedded
in a linear dissipative external circuit modeled by an Ohmic shunt resistor
$R_{S}$. This situation is equally relevant for resistively shunted Josephson
junctions in the limit of large Josephson coupling energies $E_{J}$. In this
limit one has

$G^{R}_{\chi\chi}(x,x^{\prime};\omega)\approx-\frac{2\pi i\mu}{\omega+i0},$
(205)

where $\mu=R_{Q}/R_{S}$ is the shunt dimensionless conductance and
$R_{Q}=\pi/(2e^{2})$ is the "superconducting" resistance quantum. The QPS rate
then equals to

$\Gamma(\omega)=\gamma_{QPS}^{2}(2\pi
T\tau_{R})^{2\mu}e^{\frac{\omega}{2T}}\frac{\mathbb{\Gamma}\left(\mu+\frac{i\omega}{2\pi
T}\right)\mathbb{\Gamma}\left(\mu-\frac{i\omega}{2\pi T}\right)}{8\pi
T\mathbb{\Gamma}(2\mu)},$ (206)

where $\mathbb{\Gamma}(y)$ is the Euler gamma-function and $\tau_{R}^{-1}$
plays the role of effective high-energy cutoff frequency. Evaluating the
corresponding integrals in the short wire limit
$\omega,T,I\Phi_{0}\ll\tau_{R}^{-1}$ and also for $1<\mu<3/2$, we obtain

$\Gamma^{R}(\omega)={\rm
const}-i\Gamma(\omega)e^{-\frac{\omega}{2T}}\frac{\sin\left(\pi\mu+\frac{i\omega}{2T}\right)}{\cos(\pi\mu)}.$
(207)

These expressions can be simplified in certain limits. For example, by setting
$0<\mu-1\ll 1$ we get

$\Gamma(\omega)\approx\frac{\gamma_{QPS}^{2}(2\pi
T\tau_{R})^{2\mu}e^{\frac{\omega}{2T}-2{\bf
C}(\mu-1)}\sqrt{\omega^{2}+4\pi^{2}T^{2}(\mu-1)^{2}}}{16\pi
T^{2}\mathbb{\Gamma}(2\mu)\sqrt{\sin\left(\pi\mu+\frac{i\omega}{2T}\right)\sin\left(\pi\mu+\frac{i\omega}{2T}\right)}},$
(208)

where $\bf C$ is Euler-Mascheroni constant. Also the expressions for both QPS
rates get significantly simplified in the limit $|\omega|\gg T$. One has

$\Gamma(\omega)\approx\pi\gamma_{QPS}^{2}\theta(\omega)\frac{(\omega\tau_{R})^{2\mu}}{2\omega\mathbb{\Gamma}(2\mu)},$
(209) $\Gamma^{R}(\omega)\approx{\rm
const}+\pi\gamma_{QPS}^{2}\frac{|\omega\tau_{R}|^{2\mu}e^{-i\pi\mu{\rm
sign}(\omega)}}{4\omega\mathbb{\Gamma}(2\mu)\cos(\pi\mu)}.$ (210)

Accordingly, in the zero-temperature limit one finds

$\mathcal{C}_{N}(I)=\pi\gamma_{QPS}^{2}{\rm
sign}^{N}(I)\frac{\Phi_{0}^{N+2\mu-1}\tau_{R}^{2\mu}}{2\mathbb{\Gamma}(2\mu)}|I|^{2\mu-1}.$
(211)

The above results are consistent with ones previously derived for ultrasmall
Josephson junctions [69].

###  VII.4 Higher voltage cumulants

We now turn to higher voltage cumulants at non-zero frequencies. We adopt the
following definition for the $N$-th voltage cumulant:

${\mathcal{S}}_{\omega_{1},...,\omega_{N-1}}(I)=\int
dt_{1}...dt_{N-1}e^{i\omega_{1}t_{1}+...+i\omega_{N-1}t_{N-1}}(-i)^{N}\left.\frac{\delta^{N}\mathcal{W}[J]}{\delta
J(t_{N-1})...\delta J(t_{1})\delta J(0)}\right|_{J\to 0}.$ (212)

In the zero frequency limit this definition yields

${\mathcal{S}}_{\underbrace{00...0}_{N}}(I)=\mathcal{C}_{N}(I).$ (213)

Under the condition $T,\omega\ll\tau_{R}^{-1}$ or, in other words, provided
the detector immediately "feels" QPS-generated voltage fluctuations, one can
set $\tau_{R}\to 0$ and explicitly evaluate all voltage cumulants at non-zero
frequencies. In this case the cumulant generating function takes the form

$\mathcal{W}[J]\approx-\frac{i}{2}\int
dtdt^{\prime}J(t)G^{K}_{vv}(t-t^{\prime})J(t^{\prime})\\\
-\gamma_{QPS}^{2}\int\limits_{-L/2}^{L/2}dxdx^{\prime}\int
dt\int\limits^{t}dt^{\prime}\bigl{(}P(x,x^{\prime};t-t^{\prime})-P(x^{\prime},x;t^{\prime}-t)\bigr{)}\\\
\times\sin\bigl{(}I\Phi_{0}(t-t^{\prime})\bigr{)}\sin\biggl{(}\frac{\Phi_{0}J(t)}{2}\biggr{)}\cos\biggl{(}\frac{\Phi_{0}J(t^{\prime})}{2}\biggr{)}\\\
-\frac{\gamma_{QPS}^{2}}{2}\int\limits_{-L/2}^{L/2}dxdx^{\prime}\int dt\int
dt^{\prime}\bigl{(}P(x,x^{\prime};t-t^{\prime})+P(x^{\prime},x;t^{\prime}-t)\bigr{)}\\\
\times\cos\bigl{(}I\Phi_{0}(t-t^{\prime})\bigr{)}\sin\biggl{(}\frac{\Phi_{0}J(t)}{2}\biggr{)}\sin\biggl{(}\frac{\Phi_{0}J(t^{\prime})}{2}\biggr{)}.$
(214)

One can observe that the second term in Eq. (214) can only contribute to odd
cumulants, whereas the last term determines all even cumulants. After some
algebra we arrive at the following expressions for both even and odd voltage
cumulants, respectively

${\mathcal{S}}_{\omega_{1},...,\omega_{2M}}(I)=\frac{\Phi_{0}^{2M+1}}{2^{2M}(2M)!}\sum_{p\in{\rm
perm}}\sum_{m=0}^{2M}\binom{2M}{m}\\\
\times\Bigl{(}\Gamma^{R}\bigl{(}I\Phi_{0}-(-1)^{m}(\omega_{p_{1}}+...+\omega_{p_{m}})\bigr{)}\\\
-\Gamma^{R}\bigl{(}-I\Phi_{0}-(-1)^{m}(\omega_{p_{1}}+...+\omega_{p_{m}})\bigr{)}\Bigr{)}$
(215)

and

${\mathcal{S}}_{\omega_{1},...,\omega_{2M+1}}(I)=\frac{\Phi_{0}^{2M+2}}{2^{2M+1}(2M+1)!}\sum_{p\in{\rm
perm}}\sum_{m=0}^{M}\binom{2M+1}{2m+1}\Bigl{(}\Gamma\bigl{(}I\Phi_{0}+(\omega_{p_{1}}+...+\omega_{p_{2m+1}})\bigr{)}\\\
+\Gamma\bigl{(}-I\Phi_{0}+(\omega_{p_{1}}+...+\omega_{p_{2m+1}})\bigr{)}+\Gamma\bigl{(}I\Phi_{0}-(\omega_{p_{1}}+...+\omega_{p_{2m+1}})\bigr{)}+\Gamma\bigl{(}-I\Phi_{0}-(\omega_{p_{1}}+...+\omega_{p_{2m+1}})\bigr{)},\Bigr{)}.$
(216)

where the sum is taken over all permutations of frequencies.

The above results allows us to extend the relation between the voltage
cumulants and the current-voltage characteristics of our device to non-zero
frequencies. For the odd cumulants we obtain

${\mathcal{S}}_{\omega_{1},...,\omega_{2M}}(I)=\frac{\Phi_{0}^{2M+2}I}{2^{2M-1}(2M)!}\int\frac{dI^{\prime}}{2\pi
i}V(I^{\prime})\\\ \times\sum_{p\in{\rm
perm}}\sum_{m=0}^{2M}\binom{2M}{m}\frac{1}{(I^{\prime}\Phi_{0}+(-1)^{m}(\omega_{p_{1}}+...+\omega_{p_{m}})-i0)^{2}-(I\Phi_{0})^{2}},$
(217)

while the expression for the even cumulants reads

${\mathcal{S}}_{\omega_{1},...,\omega_{2M+1}}(I)=\frac{\Phi_{0}^{2M+1}}{2^{2M+1}(2M+1)!}\sum_{p\in{\rm
perm}}\sum_{m=0}^{M}\binom{2M+1}{2m+1}\\\
\times\Biggl{(}\coth\biggl{(}\frac{I\Phi_{0}+(\omega_{p_{1}}+...+\omega_{p_{2m+1}})}{2T}\biggr{)}V\biggl{(}I+\frac{\omega_{p_{1}}+...+\omega_{p_{2m+1}}}{\Phi_{0}}\biggr{)}\\\
+\coth\biggl{(}\frac{I\Phi_{0}-(\omega_{p_{1}}+...+\omega_{p_{2m+1}})}{2T}\biggr{)}V\biggl{(}I-\frac{\omega_{p_{1}}+...+\omega_{p_{2m+1}}}{\Phi_{0}}\biggr{)}\Biggr{)}.$
(218)

These expressions can be evaluated numerically with the aid of Eqs.
(206)-(210) for $\Gamma(\omega)$ and $\Gamma^{R}(\omega)$ derived above. The
corresponding results for the third voltage cumulant as a function of two
frequencies are displayed in Fig. 12 at low and high temperatures. The third
voltage cumulant generally consists of real and imaginary parts

${\mathcal{S}}_{\omega_{1},\omega_{2}}(I)={\rm
Re}{\mathcal{S}}_{\omega_{1},\omega_{2}}(I)+i{\rm
Im}{\mathcal{S}}_{\omega_{1},\omega_{2}}(I).$ (219)

As could be seen in the plots, both these functions become considerably
smoother at higher $T$.

Figure 12: Real and imaginary parts of the third voltage cumulant at $T\to 0$
(left panels) and $T=I\Phi_{0}$ (right panels) for $\mu=1.1$.

To conclude our analysis of voltage fluctuation statistics in superconducting
nanowires we emphasize again that in the zero-frequency limit the this
statistics reduces to Poissonian similarly to the situation encountered in a
number of other tunneling-like problems. In this limit all (symmeterized)
cumulants of the voltage operator can be expressed in a simple manner through
the current-voltage characteristics of the system $V(I)$, cf. Eqs. (196),
(197). At non-zero frequencies, quantum voltage fluctuations in
superconducting nanowires are not anymore described by Poisson statistics.
This is because inter-QPS interaction produced by an effective environment
(due to the wire itself and/or an external dissipative circuit) starts playing
a more important role at shorter time scales making the whole problem much
more involved. Remarkably, also in this case it is possible to establish a
relation between the voltage cumulants and the current-voltage characteristics
of our device $V(I)$, though in a much more complicated form as compared to
that in the zero frequency limit, cf. Eqs. (217), (218). The latter
observation could be important for possible experimental verification of the
above results.

##  VIII Topology-controlled phase coherence in superconducting nanowires

It is by now well established that at $T\to 0$ a long superconducting nanowire
suffers a quantum phase transition (QPT) controlled by the wire cross section
$s$ or, equivalently by the parameter $\lambda\equiv g/8$ which we already
introduced above and which sets the magnitude of (logarithmic in space-time)
interaction between different quantum phase slips [12]. For $\lambda>2$ this
interaction is strong enough and close QPS-anti-QPS pairs are formed in the
wire which then demonstrates vanishing linear resistance $R\propto
T^{2\lambda-3}$, cf. Eqs. (155), (156). Hence, as long as $\lambda>2$ (or,
equivalently, $g>16$) the ground state of the system can be considered
superconducting. In contrast, for $\lambda<2$ inter-QPS interaction is weak,
quantum phase slips are unbound, and the wire acquires non-zero resistance
which tends to increase with decreasing $T$. The latter feature allows one to
call the wire behavior insulating provided $\lambda<2$. Thus, at $g=16$ and
$T\to 0$ one expects a superconductor-to-insulator quantum phase transition
(SIT) to occur in the systems under consideration.

Let us emphasize that possible insulating behavior of superconducting
nanowires is essentially linked to a certain type of experiment performed with
such nanowires and may not always be realized. For example, an ultrathin
superconducting nanowire forming a closed ring does not loose the ability to
carry supercurrent even for $\lambda<2$, as we have already demonstrated in
Sec. V. In this case a characteristic length scale (121) emerges beyond which
phase coherence (and, hence, supercurrent) gets exponentially suppressed by
QPS. In what follows

$R_{c}\sim L_{c}\propto\exp\left(\frac{ag_{\xi}}{2-\lambda}\right)$ (220)

emerges beyond which phase coherence (and, hence, supercurrent) gets
exponentially suppressed by QPS. The correlation length (220) diverges at
$\lambda\to 2$, thus signaling the transition to the ordered phase $\lambda>2$
with bound QPS-anti-QPS pairs and more robust superconductivity.

Figure 13: Bulk open superconducting ring attached to a superconducting
nanowire via two ultrasmall tunnel barriers located at a distance $L$ from
each other. The ring is threaded by the magnetic flux $\Phi$.

On the other hand, a closed ring geometry seriously restricts the space for
phase fluctuations, thereby enhancing the tendency towards superconductivity,
see, e.g., Ref. [70] for further discussion of this point. For this reason it
is highly desirable to analyze ground state properties of superconducting
nanowires where no fluctuation configurations are suppressed by geometry
constraints and/or boundary conditions. An example of the system of that kind
is depicted in Fig. 13. A long superconducting nanowire with sufficiently
small cross section $s$ geometric capacitance (per length) $C$ and kinetic
inductance (times length) $\mathcal{L}_{\rm kin}$ is attached to two big
superconducting reservoirs at its ends. In order to probe fluctuation effects
inside the wire it is connected to a bulk superconductor forming an open ring
by two identical small area tunnel junctions with Josephson energy $E_{J}$
located at a distance $L$ from each other at the points $x=0$ and $x=L$.
External magnetic flux $\Phi$ piercing the ring controls the phase difference
$\phi=2\pi\Phi/\Phi_{0}$ between the bulk sides of the point contacts. As
desired, within this setup fluctuations of the superconducting phase
$\varphi(x,\tau)$ remain unrestricted in any point $x$ of the wire.

Below we will analyze the effect of these fluctuations on the supercurrent
$I(\phi)$ flowing through the wire segment of length $L$ between the two
Josephson contacts [71]. As $I(\phi)$ is a $2\pi$-periodic function of $\phi$
in what follows it suffices to restrict the phase interval to $|\phi|\leq\pi$.
Below we will demonstrate that – in the setup displayed Fig. 13 – a
"disordered" phase $\lambda<2$ (or $g<16$) for a superconducting nanowire
itself consists of two different phases: A non-superconducting one with $g<2$
as well as a "mixed" one with $2<g<16$ characterized by two different
correlation lengths, $L_{c}$ (220) and

$L^{*}\propto g^{\frac{1}{1-2/g}},\quad g\geq 2.$ (221)

The latter phase is characterized by a non-trivial interplay between
supercurrent and quantum fluctuations resulting in superconducting behavior of
the wire at shorter length scales combined with its vanishing superconducting
response in the long scale limit.

###  VIII.1 Effective action

Low energy processes in the system under consideration can be described by the
effective action

$S[\varphi]=S_{TL}[\varphi]+S_{J}[\varphi(0),\varphi(L)],$ (222)

where, as before,

$S_{TL}[\varphi]=\frac{C}{8e^{2}}\int\limits_{0}^{1/T}d\tau\int
dx\left[(\partial_{\tau}\varphi)^{2}+v^{2}(\partial_{x}\varphi)^{2}\right]$
(223)

is the low energy effective action for a superconducting nanowire in the
$\varphi$-representation [12, 14, 13] and

$S_{J}[\varphi_{1},\varphi_{2}]=-E_{J}\int\limits_{0}^{1/T}d\tau\bigl{[}\cos(\varphi_{1}+\phi/2)+\cos(\varphi_{2}-\phi/2)\bigr{]}$
(224)

accounts for the Josephson energy of the contacts, where we set
$\varphi_{1}=\varphi(0,\tau)$, $\varphi_{2}=\varphi(L,\tau)$. For simplicity,
in Eq. (224)) we do not include the charging energy of the point contacts
which could always be absorbed into the first term of the wire action (223).

As the wire action (223) is Gaussian it is possible to exactly integrate out
the phase variable $\varphi(x)$ at all values $x$ except for $x=0,L$. Then we
arrive at the reduced effective action $S_{R}$ which depends on the two phase
variables $\varphi_{1}$ and $\varphi_{2}$. The grand partition function
$\mathcal{Z}$ reads

$\displaystyle\mathcal{Z}=\int D\varphi(x)\,{\rm
e}^{-S_{TL}[\varphi(x)]-S_{J}[\varphi(0),\varphi(L)]}=\int
D\varphi_{1}D\varphi_{2}\,{\rm
e}^{-S_{R}[\varphi_{1},\varphi_{2}]-S_{J}[\varphi_{1},\varphi_{2}]},$

where

$S_{R}[\varphi_{1},\varphi_{2}]=\frac{1}{2}\,\mathrm{tr}\left[\begin{pmatrix}\varphi_{1}&\varphi_{2}\end{pmatrix}\begin{pmatrix}G_{0}(0)&G_{0}(L)\\\
G_{0}(L)&G_{0}(0)\end{pmatrix}^{-1}\begin{pmatrix}\varphi_{1}\\\
\varphi_{2}\end{pmatrix}\right].$ (225)

Here the trace also includes integration over the imaginary time. The Green
function $G_{0}=\langle\varphi(x,\tau)\varphi(0,0)\rangle_{S_{0}}$ has the
form

$G_{0}(\omega_{n},x)=\frac{4e^{2}}{C}\int\frac{dq}{2\pi}\,\frac{{\rm
e}^{iqx}}{\omega_{n}^{2}+v^{2}q^{2}}=\frac{4\pi}{g|\omega_{n}|}{\rm
e}^{-\left|\frac{\omega_{n}x}{v}\right|}.$ (226)

In order to diagonalize the quadratic part of the action it is convenient to
express $S_{R}$ in terms of the variables
$\varphi_{\pm}=(\varphi_{1}\pm\varphi_{2})/2$. Then we obtain

$\displaystyle
S_{R}+S_{J}=\frac{1}{2}\sum\limits_{a=\pm}\,\mathrm{tr}\Bigl{[}\varphi_{a}G_{0,a}^{-1}\varphi_{a}\Bigr{]}-2E_{J}\int\limits_{0}^{1/T}d\tau\,\cos(\varphi_{+})\cos(\varphi_{-}-\phi/2)$

with the propagators

$G_{0,\pm}(\omega_{n})=\frac{2\pi}{g|\omega_{n}|}\left(1\pm{\rm
e}^{-\left|\frac{\omega_{n}L}{v}\right|}\right).$ (227)

Let us stress that the phase variables $\varphi_{+}$ and $\varphi_{-}$ account
for different physics in our problem. The phase
$\varphi_{-}=(\varphi(L)-\varphi(0))/2$ determines the supercurrent flowing
in-between two contacts inside the wire segment of length $L$. Hence,
configurations with non-zero $\varphi_{-}$ have non-zero energies due the
kinetic inductance of the wire and the mode corresponding to $\varphi_{-}$ has
a mass equal to $gv/2\pi L$. The variable $\varphi_{+}$, in contrast,
describes simultaneous shifts of both phases $\varphi(0)$ and $\varphi(L)$ by
the same value without producing any phase gradient along the wire. Thus, in
the absence of interactions the mode corresponding to $\varphi_{+}$ is
massless. At the same time, below we will observe that fluctuations of
$\varphi_{+}$ yield renormalization of the Josephson coupling energies $E_{J}$
of the contacts and, as such, should also be taken into account.

###  VIII.2 Variational analysis and propagators

Let us make use of the variational perturbation theory as described, e.g., in
Ref. [72]. The main idea here is to improve the standard perturbation
expansion by adding an extra term $\delta S$ which depends on the variational
parameters to the quadratic part of the action $S_{R}$. In order to accomplish
this goal the partition function (VIII.1) can be identically rewritten as

$\displaystyle\mathcal{Z}$ $\displaystyle=\int D\varphi_{1}D\varphi_{2}\,{\rm
e}^{-S_{\rm tr}}{\rm e}^{-(S_{J}-\delta S)}$ (228)

with the trial action $S_{\rm tr}=S_{R}+\delta S$. The last exponent can then
be conveniently expanded in powers of $S_{J}-\delta S$. Being expanded to all
orders, the partition function (228) obviously remains independent of the
choice of $\delta S$ and the variational parameters. However, such a
dependence emerges as long as only a finite number of terms of this expansion
is kept. Then the most accurate approximation is achieved by minimizing the
result of the perturbative expansion with respect to the variational
parameters.

In what follows we choose the trial action in the form

$\displaystyle S_{\rm
tr}=\frac{1}{2}\,\mathrm{tr}\left[\varphi_{+}(G_{0+}^{-1}+m_{+})\varphi_{+}\right]+\frac{1}{2}\,\mathrm{tr}\left[(\varphi_{-}-\psi)(G_{0-}^{-1}+m_{-})(\varphi_{-}-\psi)\right],$
(229)

which corresponds to effectively performing a self-consistent harmonic
approximation (SCHA).

Here the parameters $m_{\pm}$ represent the interaction-generated effective
masses for the modes associated with the phase variables $\varphi_{\pm}$. The
parameter $\psi$ accounts for the average value of the combination
$(\varphi(L)-\varphi(0))/2$. We note that a somewhat similar variational
calculation with a massive term was proposed in the context of Brownian motion
of a quantum particle in a periodic potential with linear Ohmic dissipation
[73]. The results obtained within the framework of this variational approach
agree with those derived by means of more rigorous techniques [41].

Expanding the last exponent in Eq. (228) in powers of $S_{J}-\delta S$ and
evaluating the integrals, for the free energy
$\mathcal{F}=-T\mathrm{ln}\mathcal{Z}$ we get

$\mathcal{F}=\mathcal{F}_{0}+\mathcal{F}_{1}+{\rm higher\;order\;terms},$
(230)

where

$\displaystyle\mathcal{F}_{0}=\frac{T}{2}\left(\,\mathrm{tr}\,\mathrm{ln}\,G^{-1}_{+}+\,\mathrm{tr}\,\mathrm{ln}\,G^{-1}_{-}\right),\quad\mathcal{F}_{1}=\bigl{\langle}S_{\rm
int}-\delta S\bigr{\rangle}_{\rm tr}.$

Neglecting all higher order terms in the expansion (230) and evaluating the
average in Eq. (VIII.2) with respect to $S_{\rm tr}$ (229), we obtain

$\mathcal{F}_{1}=-\frac{m_{+}}{2}G_{+}(0)-\frac{m_{-}}{2}G_{-}(0)+\frac{1}{2}\psi\,G_{0-}^{-1}(\omega_{n}=0)\,\psi\\\
-2E_{J}\cos(\psi-\phi/2){\rm e}^{-\bigl{(}G_{+}(0)+G_{-}(0)\bigr{)}/2},$ (231)

where $G_{\pm}^{-1}=G_{0,\pm}^{-1}+m_{\pm}$ and
$G_{\pm}(0)=T\sum\limits_{\omega_{n}}G_{\pm}(\omega_{n})$ with $\omega_{n}=\pi
T(2n+1)$ being the Matsubara frequency.

Evaluating variational derivatives of $\mathcal{F}$ with respect to $m_{\pm}$
and $\psi$, one readily finds

$\displaystyle\frac{\delta\mathcal{F}_{0}}{\delta m_{\pm}}=G_{\pm}(0)/2,$
(232) $\displaystyle\frac{\delta\mathcal{F}_{0}}{\delta\psi}=0,$ (233)
$\displaystyle\frac{\delta\mathcal{F}_{1}}{\delta
m_{\pm}}=-G_{\pm}(0)/2-\frac{1}{2}\frac{\delta G_{\pm}(0)}{\delta
m_{\pm}}\left(m_{\pm}-2E_{J}\cos(\psi-\phi/2){\rm
e}^{-\bigl{(}G_{+}(0)+G_{-}(0)\bigr{)}/2}\right),$ (234)
$\displaystyle\frac{\delta\mathcal{F}_{1}}{\delta\psi}=2E_{J}\sin(\psi-\phi/2){\rm
e}^{-\bigl{(}G_{+}(0)+G_{-}(0)\bigr{)}/2}+G_{0-}^{-1}(\omega_{n}=0)\psi.$
(235)

Imposing the extremum conditions $\delta\mathcal{F}/\delta
m_{\pm}=\delta\mathcal{F}/\delta\psi=0$ and making use of Eqs. (232)-(235) we
arrive at the following set of SCHA equations:

$m_{+}=m_{-}\equiv m$ (236)

and

$\displaystyle 2E_{J}\cos(\psi-\phi/2){\rm
e}^{-\bigl{(}G_{+}(0)+G_{-}(0)\bigr{)}/2}-m=0,$ (237) $\displaystyle
2E_{J}\sin(\psi-\phi/2){\rm
e}^{-\bigl{(}G_{+}(0)+G_{-}(0)\bigr{)}/2}+\frac{gv}{2\pi L}\psi=0.$ (238)

Note that the mass parameters $m_{\pm}$ in Eq. (236) remain equal to each
other since the two Josephson junctions involved in our problem are identical.
Equation (237) provides the relation between the effective mass $m$ and the
fluctuation-induced renormalization of the Josephson coupling energy $E_{J}$.
Equation (238) represents the equation of motion for $\psi$. It coincides with
the equation of motion for the phase $\varphi_{-}$ with $E_{J}$ renormalized
by quantum fluctuations.

As we already pointed out, the wire effective action in the form (223) applied
in the low energy limit, i.e. for $\omega,vq\ll\Delta$. Hence, a proper
ultraviolet cutoff should be imposed which respects both causality and the
fluctuation-dissipation relation. This goal is achieved by modifying the
spectral density

$J_{\pm}(\omega)=-\frac{1}{\pi}{\rm Im}\,G^{R}_{\pm}(\omega)$

making it decay at $\omega>\Delta$. The retarded Green function
$G_{\pm}^{R}(\omega)$ can be obtained from its Matsubara counterpart by means
of the standard analytic continuation procedure

$G_{\pm}^{R}(\omega)=-G_{\pm}(i\omega_{n})|_{i\omega_{n}\rightarrow\omega+i0},\;\;\;\omega_{n}>0.$

Then the Matsubara frequency summation in $G_{\pm}(0)$ can be performed with
the aid of the contour integration in the complex plane.

Employing our regularization procedure we obtain

$\displaystyle G_{\pm}(0)$
$\displaystyle=T\sum\limits_{\omega_{n}}G_{\pm}(\omega_{n})=\frac{2\pi}{g}T\sum\limits_{\omega_{n}}\left(\frac{|\omega_{n}|}{1\pm{\rm
e}^{-|\omega_{n}L/v|}}+\mu\right)^{-1}$ $\displaystyle=\frac{1}{4\pi
i}\int\limits_{\mathcal{C}}dz\,G_{\pm}(-iz)\coth\frac{z}{2T}$
$\displaystyle=\frac{i}{4\pi}\int\limits_{-\infty}^{\infty}d\omega\Bigl{(}G^{R}_{\pm}(\omega)-G^{A}_{\pm}(\omega)\Bigr{)}\coth\frac{\omega}{2T}$
$\displaystyle=\int\limits_{0}^{\Delta}d\omega\,J_{\pm}(\omega)\coth\frac{\omega}{2T},$
(239)

where the spectral density functions $J_{\pm}(\omega)$ read

$\displaystyle J_{\pm}(\omega)=-\frac{1}{\pi}{\rm
Im}\left[\frac{2\pi}{g}\left(\frac{i\omega}{\Bigl{(}1\pm{\rm e}^{i\omega
L/v}\Bigr{)}}-\mu\right)^{-1}\right],$ (240)

and we define $\mu=2\pi m/g$. In the limit $\mu L/v\ll 1$ these expressions
reduce to

$J_{+}(\omega)=\frac{4}{g}\frac{\omega}{\omega^{2}+4\mu^{2}},\quad
J_{-}(\omega)=0.$ (241)

###  VIII.3 Quantum phase transition and supercurrent

Let us now evaluate the supercurrent $I$ flowing in the wire segment of length
$L$ in-between two Josephson junctions. This current can be defined as

$\displaystyle I$
$\displaystyle=-2eT\frac{1}{\mathcal{Z}}\frac{d\mathcal{Z}}{d\phi}=2e\frac{d\mathcal{F}}{d\phi}$
$\displaystyle=-2eE_{J}\sin(\psi-\phi/2){\rm
e}^{-\bigl{(}G_{+}(0)+G_{-}(0)\bigr{)}/2}$ $\displaystyle=\frac{gev}{2\pi
L}\psi.$ (242)

Thus, within the framework of our approach the effect of phase fluctuations is
accounted for by effective renormalization of the critical current by the
factor ${\rm e}^{-\bigl{(}G_{+}(0)+G_{-}(0)\bigr{)}/2}$.

In the zero temperature limit $T\to 0$ the solution of Eq. (237) takes the
form

$\mu=\left\\{\begin{matrix}\Delta\left(\frac{4\pi
E_{J}\cos(\psi-\phi/2)}{g\Delta}\right)^{\frac{g}{g-2}},&g>2,\\\
0,&g<2,\end{matrix}\right.$ (243)

while the renormalized equation of motion (238) can be rewritten as

$\frac{\mu L}{v}\tan(\psi-\phi/2)+\psi=0.$ (244)

We observe that for $g<2$ one has $\psi=0$ and, hence, the supercurrent $I$
inside the wire is completely suppressed by strong quantum fluctuations of the
phase. on the other hand, at bigger values of $g>2$ a non-vanishing
supercurrent $I$ can flow across the wire segment between the two Josephson
junctions.

We arrive at an important conclusion: A quantum phase transition occurs at
$g=2$ separating two different phases with non-superconducting ($g<2$) and
superconducting-like ($g>2$) behavior. This dissipative QPT belongs to the
same universality class as the so-called Schmid phase transition in
resistively shunted Josephson junctions [74, 41]. It is curious that this QPT
occurs at exactly the same value of the parameter $g$ where the
superconducting gap singularity in the local electron density of states gets
suppressed due to interaction between electrons and a dissipative bath formed
by Mooij-Schön plasmons.

Let us now focus our attention on the superconducting-like phase $g>2$ and
evaluate the supercurrent $I$ affected by quantum fluctuations of the phase
$\varphi$ inside the wire. For this purpose let us combine the solution of the
equation

$\displaystyle\frac{\Delta L}{v}\left(\frac{4\pi
E_{J}}{g\Delta}\right)^{\frac{g}{g-2}}\sin(\psi-\phi/2)[\cos(\psi-\phi/2)]^{\frac{2}{g-2}}+\psi=0$
(245)

with Eq. (242). We observe that there exists a new length scale $L^{*}$ in our
problem associated with the effective mass. Making use of the Ambegaokar-
Baratoff formula for the Josephson coupling energy $E_{J}=g_{N}\Delta/8$
(where $g_{N}$ the dimensionless normal state conductance of each tunnel
junction) we can express $L^{*}$ in the form

$L^{*}=\frac{v}{\Delta}\left(\frac{2g}{\pi g_{N}}\right)^{g/(g-2)}$ (246)

Here we are merely interested in the case of small-size tunnel junctions with
few conducting channels serving as probes aiming to disturb the
superconducting wire as little as possible. Accordingly, one typically has
$g_{N}\ll g$ and $L^{*}$ diverges at $g\to 2$ remaining much longer than the
characteristic length scale $v/\Delta$ at any value of $g>2$.

The length scale (246) separates two different fluctuation regimes. For $L\gg
L^{*}$ the wire kinetic inductance contribution remains small as compared to
that of the Josephson junctions. Then the phase difference across the wire
segment in-between the junctions does not fluctuate being equal to
$\varphi(L)-\varphi(0)=\phi$. In this case we reproduce the standard mean
field current-phase relation

$I(\phi)=\frac{gev}{4\pi L}\phi.$ (247)

In the opposite limit $L\ll L^{*}$ the renormalization of $E_{J}$ already
becomes important and phase fluctuations tend to suppress the supercurrent
flowing inside the wire. In this limit we arrive at the $L$-independent result

$I(\phi)=\frac{gev}{2\pi
L^{*}}\sin\left(\frac{\phi}{2}\right)\left[\cos\left(\frac{\phi}{2}\right)\right]^{\frac{2}{g-2}}.$
(248)

Comparing the results (247) and (248) we observe that quantum fluctuations can
strongly affect both the supercurrent magnitude and the current-phase
relation. The dependence $I(\phi)$ (248) in the presence of fluctuations
becomes smoother than in Eq. (247) and the absolute value of the supercurrent
is reduced by the factor $\sim L/L^{*}$. Additional reduction of $I$
originates from the term in the square brackets in Eq. (248): The supercurrent
gets suppressed stronger for bigger values of $\phi$ The latter effect becomes
particularly significant for $g$ sufficiently close to 2. For
$\phi\rightarrow\pi$ and any $g>2$ the supercurrent tends to zero as
$I(\phi)\propto(\pi-\phi)^{\frac{g}{g-2}}$.

We also point out that for $L$ not much smaller than $L^{*}$ the supercurrent
$I(\phi\to\pi)$ behaves somewhat differently: It vanishes only for $2<g<4$,
whereas at $g>4$ one has

$I(\phi\to\pi)\approx\frac{ge\Delta}{2\pi}\left(\frac{\Delta
L}{v}\right)^{\frac{2}{g-4}}\left(\frac{\pi
g_{N}}{2g}\right)^{\frac{g}{g-4}},$ (249)

i.e. for such values of $g$ the current-phase relation remains discontinuous
at $\phi=\pi$. The dependencies $I(\phi)$ evaluated for different values of
$g$ and $L$ are also displayed in Fig. 14.

Figure 14: The phase dependent supercurrent $I(\phi)$ (expressed in units of
$e\Delta/(2\pi)$) for $E_{J}/\Delta=0.1$ and different $g$. The upper and
lower panels correspond respectively to $\Delta L/v=20$ and to $\Delta
L/v=0.5$.

Finally, we note that the form of the current-phase relation (248) derived
here resembles that obtained for resistively shunted Josephson junctions in
the presence of quantum fluctuations of the phase [75].

###  VIII.4 Effect of QPS and localization of Cooper pairs

Let us recall that the effective action $S_{TL}$ (223) employed in the above
analysis only accounts for the effect of Gaussian fluctuations of the
superconducting phase and does not yet include quantum phase slips. In order
to describe QPS effects inside the wire it is convenient to turn to the dual
$\chi$-representation for the wire effective action $S_{\rm eff}[J(\tau)=0]$
(109) derived in Sec. V. As we already discussed, this effective action
defines the sine-Gordon model which has a QPT at $T\to 0$ and $\lambda=2$ (or
$g=16$) separating two different phases [12]. Provided $g>16$ "positive" and
"negative" quantum phase slips are bound in close "neutral" pairs which do not
disrupt phase coherence at any relevant scales exceeding the superconducting
coherence length $\xi$. For such values of $g$ QPS effects are irrelevant for
any of the above results for the supercurrent $I(\phi)$ which remain
applicable without any modifications.

On the other hand, for $g<16$ quantum phase slips are no longer bound in
pairs. In this phase relevant excitations of our sine-Gordon theory are kinks
and anti-kinks (as well as their bound states) with an effective gap in the
spectrum [76, 77]

$\tilde{\Delta}\propto\gamma_{QPS}^{\frac{1}{2-\lambda}}.$ (250)

The appearance of this gap for $g<16$ (or $\lambda<2$) gives rise to the
correlation length $L_{c}\propto 1/\tilde{\Delta}$ which, similarly to the
case of superconducting nanorings, can be evaluated by equating the
renormalized QPS amplitude at this length scale
$L_{c}\tilde{\gamma}_{QPS}(L_{c})$ (see Eq. (119)) to the inductive energy
$\Phi_{0}^{2}/(2{\mathcal{L}}_{\rm kin}L_{c})\sim\xi\Delta g_{\xi}/L_{c}$ with
some numerical prefactor. Keeping track of all relevant prefactors, in the
most relevant limit $\xi\ll v/\Delta$ we obtain

$L_{c}\approx\left(\frac{\pi}{8\sqrt{2b}}\right)^{\frac{1}{1-\lambda/2}}\xi\exp\left(\frac{ag_{\xi}}{2-\lambda}\right)\left(\frac{\xi\Delta}{v}\right)^{\frac{\lambda}{2-\lambda}},$
(251)

cf. also Eq. (121). The appearance of this fundamental length scale in our
problem is directly related to the phase-charge (or flux-charge) duality. It
can be interpreted as a result of spontaneous tunneling of magnetic fluxons
$\Phi_{0}$ back and forth across the wire, as it is illustrated in Fig. 15.
These strong quantum fluctuations of magnetic flux wipe out phase coherence at
distances $\gtrsim L_{c}$ and yield effective localization of Cooper pairs at
such length scales. Accordingly, one may interpret the energy

$\tilde{E}\sim\xi\Delta g_{\xi}/L_{c}\propto\tilde{\Delta}$ (252)

as an effective Coulomb gap for the wire segment of length $\sim L_{c}$.
Viewing our nanowire as a chain of $N\approx L/L_{c}$ independent segments one
may conclude that its total Coulomb gap in the insulating regime could be as
high as $\sim\xi\Delta g_{\xi}L/L_{c}^{2}$.

Figure 15: Localization of Cooper pairs in the ground state of a uniform
superconducting nanowire due to strong zero-point fluctuations of magnetic
flux.

In the context of a setup considered here the correlation length (251) is
essentially irrelevant for $g<2$ since in this case the supercurrent $I$ is
totally suppressed already by smooth phase fluctuations.

On the other hand, for $2<g<16$ the length scale (251) becomes important. At
such values of $g$ there exist two correlation lengths, $L^{*}$ and $L_{c}$
defined respectively in Eqs. (246) and (251). The first of these two lengths
diverges at one of the phase boundaries $g=2$ whereas the second one tends to
infinity at another phase boundary $g=16$.

Consider the situation with $L^{*}<L_{c}$, in which case there exist three
regimes. At $L<L^{*}$ the supercurrent is strongly affected only by smooth
phase fluctuations and not by quantum phase slips. This regime is accounted
for by Eq. (248). At $L^{*}<L<L_{c}$ the supercurrent is insensitive to any
kind of phase fluctuations and, hence, it is given by a simple mean field
formula (247). Finally, for $L>L_{c}$ the supercurrent gets exponentially
suppressed by quantum phase slips and similarly to Eq. (122) we have

$\displaystyle
I(\phi)\sim\frac{eg_{\xi}\Delta\sqrt{L}}{\sqrt{\xi}}\left(\frac{v}{L\Delta}\right)^{\frac{3\lambda}{4}}\exp\left(-\frac{3ag_{\xi}}{4}-\left(\frac{L}{L_{c}}\right)^{1-\lambda/2}\right)\sin(\phi).$

In practical terms, the latter regime can be regarded as non-superconducting
provided $L$ strongly exceeds $L_{c}$.

It is also possible to realize the opposite situation with $L^{*}>L_{c}$, in
particular for values of $g$ close to 2. In this case the length $L^{*}$
becomes irrelevant, and one distinguishes only two regimes: $L<L_{c}$ and
$L>L_{c}$. The first one is again superconducting with the supercurrent
$I(\phi)$ affected by smooth phase fluctuations according to Eq. (248),
whereas the second regime corresponds to exponential suppression of the
supercurrent due to proliferating QPS, cf. Eq. (VIII.4). No room for the mean
field regime (247) exists at $L^{*}>L_{c}$.

###  VIII.5 An alternative setup

As we already emphasized, the setup displayed in Fig. 13 enables one to pass
an equilibrium supercurrent across a wire segment of an arbitrary length $L$
without restricting phase fluctuations inside the wire by any means. Below we
will consider an alternative setup that allows to effectively restrict the
space available for phase fluctuations by ’’pinning’’ the superconducting
phase $\varphi$ at one point inside the wire [78]. This setup is schematically
shown in Fig. 16. As before, it includes a long and thin superconducting
nanowire with one of its ends attached to a bulk superconducting reservoir.
This reservoir has a form of an open ring whose opposite end is attached to
the wire via a small-area tunnel junction at a distance $L$ along the wire.
The open ring is pierced by an external magnetic flux $\Phi$ which controls
the phase difference $\phi=2\pi\Phi/\Phi_{0}$ between its ends. Accordingly,
the phase at the left end of the wire is pinned by the reservoir and is set
equal to zero, i.e. $\varphi(x=0)=0$.

Figure 16: A superconducting nanowire directly attached to a bulk open
superconducting ring at its left end. The ring is also attached to the wire
via a small-area tunnel junction at a distance $L$ from its left end. The open
ring is pierced by an external magnetic flux $\Phi$.

We will demonstrate that such topology controlled phase pinning severely
enhances the ability of the wire to conduct supercurrent. This effect can be
interpreted in terms of the absence of a massless mode responsible for the
destruction of superconductivity at $g<2$ in the setup of Fig. 16. Instead,
the nanowire embedded in our present setup exhibits a transition between
’’more’’ and ’’less’’ superconducting phases characterized by different types
of long-range behavior. Our observable of interest is again the supercurrent
$I(\phi)$ flowing through the wire segment of length $L$ between the left wire
end and the junction. As before, the phase $\phi$ is restricted to the
interval $\phi\in(-\pi,\pi)$.

In order to proceed we will again describe our system with the aid of the
effective action (222), where the wire contribution $S_{TL}$ (223) remains the
same, whereas the Josephson term $S_{J}$ now has a somewhat simpler form

$S_{J}[\varphi(L)]=-E_{J}\int\limits_{0}^{1/T}d\tau\cos\bigl{(}\varphi(L)-\phi\bigr{)}$
(253)

as compared to Eq. (224).

Our further analysis will be fully analogous to that already carried out above
for the setup of Fig. 13. Integrating out the phase variable $\varphi(x)$ at
all points along the wire except for its value at $x=L$ we arrive at the
reduced effective action

$S_{R}+S_{J}=\frac{1}{2}\,\mathrm{tr}\left[\varphi\,G_{0}^{-1}\,\varphi\right]-E_{J}\int\limits_{0}^{1/T}d\tau\cos(\varphi-\phi)$
(254)

with

$G_{0}(\omega_{n})=\frac{8\pi}{g\omega_{n}}\tanh\left(\frac{\omega_{n}L}{v}\right)$
(255)

We observe that fluctuations of the phase variable are massive with
$m_{0}=gv/8\pi L$. The absence of a massless mode in our setup (in contrast to
that of Fig. 13) is a direct consequence of phase pinning at $x=0$ which
prohibits uniform shifts of the phase inside the wire.

Again employing the SCHA-type of analysis we define the trial action

$S_{\rm
tr}=\frac{1}{2}\,\mathrm{tr}\left[(\varphi-\psi)(G_{0}^{-1}+m)(\varphi-\psi)\right],$
(256)

where the variational parameter $m$ accounts for the interaction-induced
effective mass for the $\varphi$-mode and $\psi$ determines the average value
of the phase difference. Evaluating the free energy of the system as a
function of these two parameters and minimizing it with respect to both $m$
and $\psi$, we arrive at the following SCHA equations

$\displaystyle E_{J}\cos(\psi-\phi){\rm e}^{-G(0)/2}-m=0,$ (257)
$\displaystyle E_{J}\sin(\psi-\phi){\rm e}^{-G(0)/2}+\frac{gv}{8\pi L}\psi=0,$
(258)

where

$G(0)=T\sum\limits_{\omega_{n}}\left(G_{0}^{-1}(\omega_{n})+m\right)^{-1}.$
(259)

As before, the effect of phase fluctuations reduces to effective
renormalization of $E_{J}$ by the factor ${\rm e}^{-G(0)/2}$. The supercurrent
$I$ is then found from the equation

$I=\frac{gev}{4\pi L}\psi.$ (260)

At $L\lesssim v/\Delta$ phase fluctuations are strongly suppressed and the
system remains in the mean-field regime. In the opposite limit of large $L$
the solution of Eq. (257) exhibits two qualitatively distinct regimes. At
$g<4$ we find $|m|\ll v/L$. Therefore, the emergent mass is negligible, and
the effect of fluctuations is purely Gaussian. The equation of motion (258) is
then rewritten as

$E_{J}\sin(\psi-\phi)\left(\frac{\Delta L}{v}\right)^{-4/g}+\frac{gv}{8\pi
L}\psi=0.$ (261)

In the interesting for us limit of small $E_{J}$ we may readily set
$\frac{8\pi E_{J}}{g\Delta}<1$. In this case the sine term is renormalized to
zero faster than the kinetic inductance contribution $\propto L^{-1}$ and,
hence, we obtain

$I(\phi)=2eE_{J}\left(\frac{v}{\Delta L}\right)^{4/g}\sin\phi.$ (262)

This expression demonstrates that for $g<4$ phase fluctuations (i) modify the
current-phase relation making it sine-like instead of the sawtooth-like and
(ii) yield a decrease of the supercurrent as compared to the standard
Josephson formula $I(\phi)=2eE_{J}\sin\phi$ that applies in the limit
$L\rightarrow 0$. In addition, we observe that in the presence of fluctuations
the supercurrent (262) decays faster with increasing $L$ than the standard
mean field dependence $I\propto 1/L$.

Let us now turn to the case $g>4$. Resolving Eq. (257) in the limit
$L\rightarrow\infty$ we obtain

$m=\left\\{\begin{matrix}\left[E_{J}\cos(\psi-\phi)\left(\frac{8\pi}{\Delta
g}\right)^{\frac{4}{g}}\right]^{\frac{g}{g-4}},&\cos(\psi-\phi)>0,\\\ -gv/8\pi
L+o(1/L),&\cos(\psi-\phi)<0.\end{matrix}\right.$ (263)

This solution remains valid only as long as $L$ exceeds the length scale

$L^{*}=\frac{v}{\Delta}\left(\frac{g}{\pi g_{N}}\right)^{\frac{g}{g-4}},$
(264)

cf. Eq. (246).

This length separates the regime $L>L^{*}$ where fluctuations yield non-
Gaussian renormalization of the interaction potential from the Gaussian regime
$L\ll L^{*}$ where $|m|\ll gv/8\pi L$. As long as $v/\Delta\ll L\ll L^{*}$ the
current is again given by Eq. (262).

For $g>4$ the renormalized Josephson coupling energy decreases slower than
$1/L$ and at $L\sim L^{*}$ it becomes of the same order as the kinetic
inductance contribution. At even larger distances mass renormalization
saturates to the value defined in Eq. (263). The kinetic inductance
contribution, on the contrary, decreases as $1/L$. Therefore, at $L\gg L^{*}$
the phase is pinned to the lowest minimum of the renormalized Josephson
junction potential, i.e. we have $\psi=\phi$. In this case the current-phase
relation reduces to the standard mean field form (247).

Finally, we note that the effects of QPS inside the superconducting nanowire
are analyzed in exactly the same manner as it was already done above in Sec.
IXD, except one should now distinguish two phases – $g<4$ and $4<g<16$
(instead of $g<2$ and $2<g<16$) – and replace the expression for $L^{*}$ (246)
by Eq. (264).

For $g<4$ there exists only one correlation length (251) in our problem. At
$L\ll L_{c}$ QPS effects are irrelevant and the supercurrent suppression is
merely due to smooth phase fluctuations. In this limit Eq. (262) applies and
the supercurrent decays as a power-law with increasing $L$. As soon as $L$
exceeds $L_{c}$ quantum phase slips come into play and the supercurrent decay
becomes exponential with $L$ according to Eq. (VIII.4).

At $4<g<16$ there already exist two different correlation lengths, $L^{*}$ and
$L_{c}$. The first one diverges as $g\rightarrow 4$ while the second one tends
to infinity at $g\rightarrow 16$. Depending on the relation between these two
lengths, different regimes can occur.

Consider, e.g., the limit $L^{*}\ll L_{c}$ which can always be realized for
sufficiently large values of $g_{\xi}$. In this case, at shorter length scales
$L<L^{*}$ only smooth phase fluctuations affect the supercurrent causing its
power-law suppression with increasing $L$ and the sinusoidal current-phase
relation, see Eq. (262). At $L^{*}<L<L_{c}$ both smooth phase fluctuations and
quantum phase slips are irrelevant and the supercurrent is defined by the
standard mean field result $I\propto 1/L$ (247) describing the sawtooth-shaped
current-phase relation. Finally, at $L\gg L_{c}$ the current is exponentially
suppressed and the current-phase relation again reduces to the sine form
(VIII.4).

###  VIII.6 Discussion

The above analysis demonstrates that superconducting properties of metallic
nanowires depend not only on their parameters, but also on the topology of the
experimental setup and on the way the experiment is being performed. The
ability of the wire to carry supercurrent also varies at different length
scales being affected by different kinds of fluctuations including, on one
hand, sound-like collective plasma modes forming a quantum dissipative
environment for electrons inside the wire and, on the other hand, quantum
phase slips. As the bath of plasma modes is (almost) Ohmic the low temperature
system behavior resembles that involving a Schmid-like dissipative QPT [74,
41] either at $g=2$ or at $g=4$ depending on the setup under consideration.
The presence of quantum phase slips naturally leads to a BKT-type quantum
phase transition at $g=16$ [12].

While the system displayed in Fig. 13 allows for unrestricted fluctuations of
the superconducting phase, the setup of Fig. 16 effectively pins the phase at
one of the wire ends. In the former case a gapless Ohmic mode (associated with
uniform phase shifts along the wire) appears at any $L$. Fluctuations
associated with this gapless mode cause a Schmid-like QPT at $g=2$. As a
result, the wire completely loses superconductivity at $g<2$, whereas the
phase with $2<g<16$ is mixed, i.e. it is non-superconducting in the long
length limit and superconducting at shorter scales, even though the gapless
mode causes additional suppression of current in the limit $L\rightarrow 0$.
Comparing this situation with the one encountered for the structure of Fig.
16, we observe that in the latter case superconductivity is severely enhanced
as a result of the soft mode suppression due to phase pinning. This effect
turns the QPT into a transition between ’’less’’ and ’’more’’ superconducting
phases at $g=4$ . Note that a similar phase transition was also discussed in
Ref. [79] in the context of superconducting nanorings interrupted by a
Josephson junction.

##  IX Quantum phase slips in capacitively coupled superconducting nanowires

Let us now extend our analysis of QPS-related effects to yet another structure
which consists of capacitively coupled superconducting nanowires. We will
demonstrate that quantum fluctuations in one of the two wires effectively "add
up" to those of another one even without any direct electric contact between
them, thereby giving rise to a number of interesting effects, such as, e.g.,
splitting of plasmon modes and interaction-induced SIT shifting in each of the
wires.

###  IX.1 The model

Consider the system of two long parallel to each other superconducting
nanowires, as it is schematically shown in Fig. 17.

Figure 17: Two capacitively coupled superconducting nanowires.

As before, the wires are described by geometric capacitances $C_{1}$ and
$C_{2}$ (per unit wire length) and kinetic inductances $\mathcal{L}_{1}$ and
$\mathcal{L}_{2}$ (times length) effectively representing the two transmission
lines. Capacitive coupling between these two nanowires is accounted for by the
mutual capacitance $C_{m}$. Generalizing our analysis in Sec. IVA to the above
structure, we arrive at the contribution to the system Hamiltonian that keeps
track of both electric and magnetic energies in these coupled transmission
lines. It reads [88, 89]

$\displaystyle\hat{H}_{TL}=\frac{1}{2}\sum_{i,j=1,2}\int
dx(\mathcal{L}^{-1}_{ij}\hat{\Phi}_{i}(x)\hat{\Phi}_{j}(x)+(1/\Phi^{2}_{0})C^{-1}_{ij}(\partial_{x}\hat{\chi}_{i}(x)\partial_{x}\hat{\chi}_{j}(x)),$
(265)

where $x$ is the coordinate along the wires, $\mathcal{L}_{ij}$ and $C_{ij}$
denote the matrix elements of the inductance and capacitance matrices

$\check{\mathcal{L}}=\left(\begin{array}[]{crl}\mathcal{L}_{1}&0\\\
0&\mathcal{L}_{2}\\\
\end{array}\right),\quad\check{C}=\left(\begin{array}[]{crl}C_{1}&C_{m}\\\
C_{m}&C_{2}\\\ \end{array}\right)$ (266)

As before, the Hamiltonian (265) is expressed in terms of the dual operators
$\hat{\chi}(x)$ and $\hat{\Phi}(x)$ which obey the canonical commutation
relation

$[\hat{\Phi}_{i}(x),\hat{\chi}_{j}(x^{\prime})]=-i\delta_{ij}\Phi_{0}\delta(x-x^{\prime}).$
(267)

Provided the wires are thick enough the low energy Hamiltonian in Eq. (265) is
sufficient. However, for thinner wires one should also account for the effect
of quantum phase slips. The corresponding contribution to the total
Hamiltonian for our system can be expressed in the form (cf., Eq. (81))

$\hat{H}_{QPS}=-\sum_{j=1,2}\gamma_{j}\int dx\cos(\hat{\chi}_{j}(x)),$ (268)

where

$\gamma_{j}\sim(g_{j\xi}\Delta/\xi)\exp(-ag_{j\xi}),\;\;\;a\sim 1,\;\;\;j=1,2$
(269)

are the QPS amplitudes per unit wire length, $\Delta$ is the superconducting
order parameter in each of the wires, $g_{j\xi}=R_{q}/R_{j\xi}$ and $R_{j\xi}$
is the normal state resistance of the $j$-th wire segment of length equal to
the superconducting coherence length $\xi$.

The total Hamiltonian for the system under consideration equals to the sum of
the two terms in Eqs. (265) and (268),

$\hat{H}=\hat{H}_{TL}+\hat{H}_{QPS},$ (270)

representing an effective sine-Gordon model that will be treated below.

###  IX.2 Plasma mode splitting

Any QPS event causes redistribution of charges inside the wire and generates a
pair of voltage pulses propagating simultaneously in the opposite directions
along the wire with the velocity $v$, see also Fig. 18 (left panel). In a
single wire this process is controlled, e.g., by the wave equation (189) for
the operator $\hat{\chi}$. In the case of two capacitively coupled wires a QPS
event in one of the wires generates voltage pulses in both wires. The
corresponding generalization of the wave equation (189) for the operators
$\hat{\chi}_{1,2}$ in each of the two wires follows directly from the system
Hamiltonian (265), and the corresponding voltage operators $\hat{V}_{1,2}$
read

$\hat{V}_{i}(t)=1/\Phi_{0}\sum_{j=1,2}C^{-1}_{ij}(\partial_{x}\hat{\chi}_{j}(x_{1},t)-\partial_{x}\hat{\chi}_{j}(x_{2},t)).$
(271)

Figure 18: Time-dependent phase configurations describing a QPS event at $t=0$
(red) and $t>0$ (blue) together with propagating voltage pulses generated by
this QPS event in a single superconducting nanowire (left panel) and in the
first of the two capacitively coupled superconducting nanowires (middle
panel).In the case of the two wires each of the voltage pulses is split into
two propagating with different velocities $v_{\pm}$. Right panel: ime-
dependent phase configurations at $t=0$ (red) and $t>0$ (blue) together with
propagating voltage pulses in the second of the two capacitively coupled
superconducting nanowires generated by a QPS event in the first one.

It will be convenient for us to go over to the phase representation and
express the corresponding equation of motion for the phases $\varphi_{1,2}$ in
both wires in the form (cf., e.g., Eq.(88))

$\left(\check{1}\partial_{t}^{2}-\check{\mathcal{V}}^{2}\partial_{x}^{2}\right)\left(\begin{array}[]{crl}\varphi_{1}(x,t)\\\
\varphi_{2}(x,t)\\\ \end{array}\right)=0,$ (272)

where $\check{\mathcal{V}}=(\check{C}\check{\mathcal{L}})^{-1/2}$ is the
velocity matrix that accounts for plasmon modes propagating along the wires.
Assume now that a QPS event occurs at the initial time moment $t=0$ at the
point $x=0$ inside the first wire. As we just discussed, at $t>0$ voltage
pulses originating from this QPS event will propagate in both wires. Resolving
Eq. (272) together with the proper initial condition corresponding to a QPS
event (cf., e.g., Eq. (89)), we arrive at the following picture.

In the first wire each of the two voltage pulses propagating in opposite
directions turns out to be split into two pulses of the same sign propagating
with different velocities $v_{+}$ and $v_{-}$, as it is also illustrated in
Fig. 18 (middle panel). Each of the voltage pulses generated in the second
wire by a QPS event in the first one is also split into two, they also
propagate with two different velocities $v_{+}$ and $v_{-}$, however the signs
of these voltage pulses are now opposite to each other, cf. Fig. 18 (right
panel). The velocities of these split plasmon modes are determined by the
eigenvalues of the velocity matrix $\check{\mathcal{V}}$. They read [90]

$v_{\pm}=\frac{1}{2\kappa}\left[\sqrt{v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\kappa}\pm\frac{\sqrt{(v_{1}^{2}-v_{2}^{2})^{2}+\frac{4C_{m}^{2}v_{1}^{2}v_{2}^{2}}{C_{1}C_{2}}}}{\sqrt{v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\kappa}}\right],$
(273)

where $\kappa=\sqrt{1-C_{m}^{2}/(C_{1}C_{2})}$ and
$v_{i}=1/\sqrt{C_{i}\mathcal{L}_{i}}$ ($i=1,2$) is the velocity of the Mooij-
Schön modes in the $i$-th wire for $C_{m}\to 0$. In the case of identical
wires with $C_{1}=C_{2}=C$ and
${\mathcal{L}}_{1}={\mathcal{L}}_{2}={\mathcal{L}}$ the result (273) reduces
to a particularly simple form

$v_{\pm}=1/\sqrt{{\mathcal{L}}(C\mp C_{m})}.$ (274)

This expression indicates that one of these velocities may strongly increase
provided the wires are located close enough to each other in which case the
cross-capacitance $C_{m}$ may become of order $C$. In the absence of inter-
wire interaction we have $\kappa=1$ and Eq. (273) obviously yields
$v_{+}=v_{1}$ and $v_{-}=v_{2}$.

###  IX.3 Quantum phase transitions: renormalization group analysis

Let us now turn to the issue of a superconductor-insulator QPT in capacitively
coupled superconducting nanowires with proliferating quantum phase slips. As
we just demonstrated, the presence of capacitive coupling between the
nanowires may significantly affect plasmon propagation in our system and,
hence, alter the interaction between quantum phase slips. In this case, SIT in
each of the wires is controlled not only by the parameters of the
corresponding wire but also by those of the neighboring one as well as by the
mutual capacitance. Interestingly enough, superconducting nanowires with
properly chosen parameters may turn insulating once they are brought
sufficiently close to each other [89].

In order to quantitatively describe QPT in coupled superconducting nanowires
we will employ the renormalization group (RG) analysis. This approach is well
developed and was successfully applied to a variety of problems in condensed
matter theory, such as, e.g., the problem of weak Coulomb blockade in tunnel
[41, 80, 81, 82] and non-tunnel [83, 84, 85] barriers between normal metals or
that of a dissipative phase transition in resistively shunted Josephson
junctions [41, 74, 86, 87]. In the case of superconducting nanowires QPT was
described [12] with the aid of RG equations equivalent to those initially
developed for two-dimensional superconducting films [11] which exhibit
classical BKT phase transition driven by temperature. In contrast, as we
already discussed above, quantum SIT in quasi-one dimensional superconducting
wires is controlled by the parameter $\lambda$ proportional to the square root
of the wire cross section $s$.

Provided the two superconducting wires depicted in Fig. 17 are decoupled from
each other, i.e. for $C_{m}\to 0$, one should expect two independent QPT to
occur in these two wires respectively at $\lambda_{1}=2$ and at
$\lambda_{2}=2$, where
$\lambda_{1,2}=(R_{q}/8)\sqrt{C_{1,2}/{\mathcal{L}}_{1,2}}$. In the presence
of capacitive coupling between the wires these two QPT get modified. In order
to account for these modifications let us express the grand partition function
of our system ${\mathcal{Z}}={\rm Tr}\exp(-\hat{H}/T)$ in terms of the path
integral

${\mathcal{Z}}=\int D\chi_{1}\int D\chi_{2}\exp(-S[\chi_{1},\chi_{2}]),$ (275)

where

$\displaystyle S=\frac{1}{2\Phi^{2}_{0}}\sum_{i,j=1,2}\int
dxd\tau\Big{(}\xi\Delta\mathcal{L}_{ij}\partial_{\tau}\chi_{i}\partial_{\tau}\chi_{j}+\frac{1}{\xi\Delta}C^{-1}_{ij}\partial_{x}\chi_{i}\partial_{x}\chi_{j}\Big{)}-\sum_{i=1,2}y_{i}\int
dxd\tau\cos(\chi_{i})$ (276)

is the effective action corresponding to the Hamiltonian (270) and
$y_{i}=\gamma_{i}\xi/\Delta$ denote the effective fugacity for the gas of
quantum phase slips in the $i$-th wire. For the sake of convenience in Eq.
(276) we rescaled the spatial coordinate in units of $x_{0}$, i.e. $x\to x\xi$
and the time coordinate in units of $\tau_{0}$, i.e. $\tau\to\tau/\Delta$.

As usually, let us divide the $\chi$-variables into fast and slow components
$\chi_{i}=\chi_{i}^{f}+\chi_{i}^{s}$, where

$\displaystyle\chi^{f}_{i}(x,\tau)=\int_{\Lambda<\omega^{2}+q^{2}<\Lambda+\delta\Lambda}\frac{d\omega
dq}{2\pi}\chi_{\omega,q}e^{i\omega\tau+iqx},$
$\displaystyle\chi^{s}_{i}(x,\tau)=\int_{\omega^{2}+q^{2}<\Lambda}\frac{d\omega
dq}{2\pi}\chi_{\omega,q}e^{i\omega\tau+iqx}.$

Assuming $\delta\Lambda/\Lambda\ll 1$, expanding in $\chi_{i}^{f}$ and
integrating these fast variables out we employ the perturbation theory in
$y_{1,2}$ and observe that in order to account for the leading order
corrections it suffices to evaluate the following matrix Green function at
coincident points

$\check{G}^{f}(0,0)=\Phi^{2}_{0}\int\frac{d\omega
dq}{(2\pi)^{2}}\Big{(}\xi\Delta\check{\mathcal{L}}\omega^{2}+\frac{1}{\xi\Delta}\check{C}^{-1}q^{2}\Big{)}^{-1}=2(\delta\Lambda/\Lambda)\check{\lambda},$
(277)

where $\check{\lambda}=(R_{q}/8)\check{\mathcal{V}}\check{C}$. The matrix
$\check{\lambda}$ reads

$\check{\lambda}=\frac{1}{\sqrt{\frac{1}{v^{2}_{1}}+\frac{1}{v^{2}_{2}}+\frac{2\kappa}{v_{1}v_{2}}}}\left(\begin{array}[]{crl}\lambda_{1}\left(\frac{1}{v_{1}}+\frac{\kappa}{v_{2}}\right)&R_{q}C_{m}/8\\\
R_{q}C_{m}/8&\lambda_{2}\left(\frac{1}{v_{2}}+\frac{\kappa}{v_{1}}\right)\\\
\end{array}\right).$ (278)

Following the standard analysis [11] and proceeding to bigger and bigger
scales $\Lambda$, we eventually arrive at the following RG equations for the
QPS fugacities $y_{1}$ and $y_{2}$:

$\frac{dy_{i}}{d\log\Lambda}=(2-\lambda_{ii})y_{i},\quad i=1,2,$ (279)

where $\lambda_{11}$ and $\lambda_{22}$ are diagonal elements of the matrix
$\check{\lambda}$. Note that here we restrict our RG analysis to the lowest
order in $y_{1,2}$, in which case other parameters of our problem remain
unrenormalized.

It follows immediately from Eqs. (279) that our system exhibits two BKT-like
QPT at $\lambda_{11}=2$ and $\lambda_{22}=2$. For the first wire the
corresponding phase transition point is fixed by the condition [89]

$\lambda_{1}=2\frac{\sqrt{1+\frac{v^{2}_{1}}{v^{2}_{2}}+2\kappa\frac{v_{1}}{v_{2}}}}{1+\kappa\frac{v_{1}}{v_{2}}}.$
(280)

The same condition for the second wire is derived from Eq. (280) simply by
interchanging the indices $1\leftrightarrow 2$.

These results allow to conclude that in the presence of capacitive coupling
quantum fluctuations in one of these wires tend to decrease superconducting
properties of the other one. As a result, SIT in both wires occurs at larger
values of $\lambda_{1,2}$ than in the absence of such coupling.

Equation (280) demonstrates that the magnitude of this effect depends on the
ratio of the plasmon velocities in the two wires $v_{1}/v_{2}$ and on the
strength of the capacitive coupling controlled by $C_{m}$. Provided the wire
cross sections $s_{1}$ and $s_{2}$ differ strongly the plasmon velocities
$v_{i}\propto\sqrt{s_{i}}$ also differ considerably. Provided, e.g., the first
wire is much thinner than the second one we have $v_{1}\ll v_{2}$ and, hence,
the QPT condition (280) in the first wire remains almost unaffected for any
capacitive coupling strength. If, on the contrary, the first wire is much
thicker than the second one, then one has $v_{1}\gg v_{2}$ and the condition
(280) reduces to $\lambda_{1}\simeq 2/\sqrt{1-C_{m}^{2}/(C_{1}C_{2})}$
implying that the critical value $\lambda_{1}$ can exceed 2 considerably for
sufficiently large values $C_{m}$.

Figure 19: Top: Critical surfaces corresponding to SIT at $\lambda_{11}=2$ and
$\lambda_{22}$=2. Bottom: Phase diagram for two capacitively coupled
superconducting nanowires with $\lambda_{1}=2.01$ and $\lambda_{2}=2.03$. Both
curves $\lambda_{11}(C_{m})$ and $\lambda_{22}(C_{m})$ decrease and cross the
critical line $\lambda_{c}=2$ with increasing mutual capacitance $C_{m}$.

It is fairly obvious that capacitive coupling depends on the distance between
the wires. While at large distances this coupling is negligible, as the wires
get closer to each other the value $C_{m}$ increases and, hence, their mutual
influence increases as well. Let us choose the wire parameters in such a way
that for $C_{m}=0$ both $\lambda_{1}$ and $\lambda_{2}$ slightly exceed 2,
i.e. the wires remain in the superconducting phase being relatively close to
SIT. Moving the wires closer to each other we "turn on" capacitive coupling
between them and, hence, decrease both values $\lambda_{1}$ and $\lambda_{2}$
below 2. As a result, two superconducting wires become insulating as soon as
they are brought sufficiently close to each other. This remarkable physical
phenomenon is illustrated by the phase diagram in Fig. 19.

We can also add that transport properties can be investigated in exactly the
same manner as in the case of a single nanowire. Generalization of the
technique [12] is straightforward and yields

$R_{i}(T)\propto\gamma^{2}_{i}T^{2\lambda_{ii}-3},\quad i=1,2,$ (281)

where $R_{i}(T)$ is the linear resistance of the $i$-th wire. This result
remains applicable either for $\lambda_{ii}>2$ or for any $\lambda_{ii}$ at
sufficiently high temperatures.

The effects discussed here can be observed in a variety of structures
involving superconducting nanowires. For instance, superconducting nanowires
in the form of a meander are quite frequently employed in experiments, see,
e.g., Ref. [91]. Different segments of such a wire remain parallel to each
other being close enough to develop electromagnetic coupling. One can expect,
therefore, that the wire of such a geometry should be "less superconducting"
than the same wire that has the form of a straight line.

Let us first mimic the behavior of the wire depicted in Fig. 20 by considering
three identical parallel to each other capacitively coupled superconducting
nanowires. For simplicity we will assume the nearest neighbor interaction,
i.e. the second (central) nanowire is coupled to both the first and the third
nanowires via the mutual capacitance $C_{m}$ whereas the latter two are
decoupled from each other. Quantum properties of this system are described by
the same effective action (276) where the inductance and capacitance matrices
now read

$\check{\mathcal{L}}=\left(\begin{array}[]{crl}\mathcal{L}&0&0\\\
0&\mathcal{L}&0\\\ 0&0&\mathcal{L}\\\
\end{array}\right),\quad\check{C}=\left(\begin{array}[]{crl}C&C_{m}&0\\\
C_{m}&C&C_{m}\\\ 0&C_{m}&C\\\ \end{array}\right),$ (282)

and the summation runs over the indices $i,j=1,2,3$. Proceeding along the same
lines as above we again arrive at Eq. (277), where the diagonal elements of
the matrix $\check{\lambda}$ now read [89]

$\displaystyle\lambda_{22}=\frac{\lambda}{2}\left(\sqrt{1-\frac{\sqrt{2}C_{m}}{C}}+\sqrt{1+\frac{\sqrt{2}C_{m}}{C}}\right),$
(283)

$\lambda_{11}=\lambda_{33}=\lambda/2+\lambda_{22}/4$ and $\lambda=g/8$ with
$g$ defined in Eq. (3). We again recover the RG equations (279), now with
$i=1,2,3$. Being combined with Eq. (283), these RG equations demonstrate that
in the presence of capacitive coupling between nanowires SET in the central
nanowire occurs at $\lambda_{22}=2$ implying $\lambda>2$ similarly to the case
of two coupled nanowires.

Figure 20: A superconducting nanowire in the form of a meander.

The RG equation (279) with $i=2$ combined with Eq. (283) also accounts for QPT
in the wire having the form of a meander displayed in Fig. 20. In this case,
within the approximation of the nearest neighbor capacitive interaction
between the wire segments QPT occurs at

$\lambda=\frac{4}{\sqrt{1-\sqrt{2}\frac{C_{m}}{C}}+\sqrt{1+\sqrt{2}\frac{C_{m}}{C}}},$
(284)

i.e. the critical value of the parameter $\lambda$ exceeds 2 for any
$C_{m}\neq 0$. The approximation of the nearest neighbor interaction appears
to be well justified in the limit $C_{m}\ll C$. For stronger interactions with
$C_{m}\sim C$ this approximation becomes insufficient for a quantitative
analysis. Qualitatively, our key observations should remain applicable also in
this case: A nanowire in the form of a straight line with $\lambda$ slightly
exceeding the critical value 2 should demonstrate superconducting-like
behavior with $R(T)\propto T^{2\lambda-3}$ [12] whereas the wire with exactly
the same parameters may turn insulating provided it has the form of a meander
with capacitive coupling between its segments.

##  X Concluding remarks

In this review we have made an effort to cover a number of recent developments
related to quantum properties of superconducting nanowires putting an emphasis
on fundamental aspects of the theory of such systems. Unlike in the case of
bulk superconductors, the low temperature physics of quasi-one-dimensional
nanowires and nanorings is essentially determined by quantum fluctuations
which are, in turn, controlled by two different parameters, the dimensionless
normal state conductance of the wire segment $g_{\xi}$ (2) and the
dimensionless wire admittance $g$ (3). Both these parameters decrease with
decreasing wire cross section making quantum fluctuations progressively more
pronounced.

Provided the parameter $g_{\xi}$ remains very large, the low temperature
behavior of the system is determined by small (Gaussian) quantum fluctuations
of the phase of the order parameter which – for not very large values of $g$ –
significantly affect the electron density of states in superconducting
nanowires and cause supercurrent noise in superconducting nanorings. As soon
as the dimensionless conductance $g_{\xi}$ becomes not too large quantum phase
slips come into play. As a result, current-biased superconducting nanowires
acquire a non-zero resistance and exhibit shot noise of the voltage. These
phenomena can be conveniently interpreted in terms of tunneling of quantum
fluxons (i.e. the flux quanta $\Phi_{0}$) across the nanowire. In the context
of phase-charge duality, these fluxons can be treated as effective quantum
"particles" exactly dual to Cooper pairs with charge $2e$. Such "particles"
obey complicated full counting statistics which, however, reduces to
Poissonian one in the zero frequency limit.

Quantum phase slips may strongly affect both the supercurrent and its
fluctuations in superconducting nanorings. This effect can be particularly
pronounced if QPS remain unbound, i.e. for $g<16$ or, equivalently, for
$\lambda<2$. In this case and provided the ring radius exceeds the critical
value $R_{c}$ (121), even at $T\to 0$ strong quantum fluctuations essentially
"dephase" and suppress supercurrent that could flow across the ring.

The same non-perturbative length scale $L_{c}$ (251) emerges in
superconducting nanowires. For $g<16$ and $T\to 0$ such nanowires do show an
insulating behavior at scales exceeding typical size of a "superconducting
domain" $L_{c}$, whereas at shorter length scales they may exhibit
superconducting properties albeit possibly affected by quantum fluctuations of
the phase. Hence, $L_{c}$ can be interpreted as localization length for Cooper
pairs.

Direct experimental evidence for this kind of behavior and for the presence of
this localization length was very recently observed in nominally uniform
titanium nanowires [92]. Moreover, reanalyzing similar data reported
previously [24, 25, 93] for a large number of $MoGe$ nanowires we conclude
that these data are also consistent with the above physical picture involving
the correlation length $L_{c}$ (251), i.e. the superconducting $MoGe$ samples
[24, 25, 93] obey the condition $L\lesssim L_{c}$, whereas the non-
superconducting ones typically have the length $L$ exceeding $L_{c}$. It is
also important to emphasize that the nanowires employed in all these
experiments did not contain any grains or dielectric barriers. Hence,
similarly to normal metallic structures [1] these observations can be
interpreted as a manifestation of weak Coulomb blockade of Cooper pairs that
may occur even in the absence of any tunnel barriers.

Further non-trivial properties of superconducting nanowires may be sensitive
to specific topology of the experimental setup under consideration. A number
of interesting phenomena associated with quantum phase slips also occurs in
capacitively coupled superconducting nanowires.

Finally, it is worthwhile to point out that non-trivial quantum properties of
superconducting nanowires and nanorings open up plenty of possibilities for
their applications in nanoelectronics, metrology and quantum information
technology. Various devices, such as, e.g., single-charge transistor [48] and
charge quantum interference device [49] have already been demonstrated.
Superconducting nanowires can also be employed as central elements for QPS
flux qubits [53] as well as for creating a QPS-based standard of electric
current [50] and single photon detectors [91]. We are confident that intensive
investigations of intriguing fundamental properties of quasi-one-dimensional
superconducting structures as well as their technological applications will
continue in the near future.

Acknowledgements

We would like to thank K.Yu. Arutyunov, A. Radkevich and A. Latyshev for
stimulating discussions and collaboration on a number of issues touched upon
in this review. Our work was supported by RFBR grant 19-12-50260.

## References

  * [1] A.D. Zaikin and D.S. Golubev, Dissipative Quantum Mechanics of Nanostructures: Electron Transport, Fluctuations and Interactions (Jenny Stanford, Singapore, 2019).
  * [2] K.Yu. Arutyunov, D.S. Golubev, and A.D. Zaikin, Phys. Rep. 464, 1 (2008).
  * [3] A.I. Larkin and A.A. Varlamov, Theory of fluctuations in superconductors (Clarendon, Oxford, 2005).
  * [4] A. Bezryadin, J. Phys.: Condens. Matter 20, 043202 (2008).
  * [5] A.D. Zaikin, in: Handbook of Nanophysics: Nanotubes and Nanowires (CRC Press, Boca Raton, FL, 2010) p. 40-1.
  * [6] A. Bezryadin, Superconductivity in Nanowires (Wiley-VCH, Weinheim, 2013).
  * [7] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).
  * [8] P.C. Hohenberg, Phys. Rev. 158, 383 (1967).
  * [9] V.S. Berezinskii, Sov. Phys. JETP 32 493; 34, 610 (1971).
  * [10] J.M. Kosterlitz and D.J. Thouless, J. Phys. C 6, 1181 (1973).
  * [11] J.M. Kosterlitz, J. Phys. C 7, 1046 (1974).
  * [12] A.D. Zaikin, D.S. Golubev, A. van Otterlo, and G.T. Zimanyi, Phys. Rev. Lett. 78, 1552 (1997).
  * [13] A. van Otterlo, D.S.Golubev, A.D.Zaikin, and G.Blatter, Eur. Phys. J. B 10, 131 (1999).
  * [14] D.S. Golubev and A.D. Zaikin, Phys. Rev. B 64, 014504 (2001).
  * [15] D.J. Scalapino, S.R. White, and S. Zhang, Phys. Rev. B 47, 7995 (1993).
  * [16] R. Resta, J. Phys.: Condens. Matter 30, 414001 (2018).
  * [17] D.S. Golubev and A.D. Zaikin, Phys. Rev. B 78, 144502 (2008).
  * [18] J.E. Mooij and G. Schön, Phys. Rev. Lett. 55, 114 (1985).
  * [19] W.A. Little, Phys. Rev. 156, 396 (1967).
  * [20] J.S. Langer and V. Ambegaokar, Phys. Rev. 164, 498 (1967).
  * [21] D.E. McCumber and B.I. Halperin, Phys. Rev. B 1 1054 (1970).
  * [22] J.E. Lukens, R.J. Warburton, and W.W. Webb, Phys. Rev. Lett. 25, 1180 (1970).
  * [23] R.S. Newbower, M.R. Beasley, and M. Tinkham, Phys. Rev. B 5, 864 (1972).
  * [24] A. Bezryadin, C.N. Lau, and M. Tinkham, Nature 404, 971 (2000).
  * [25] C.N. Lau, N. Markovic, M. Bockrath, A. Bezryadin, and M. Tinkham, Phys. Rev. Lett. 87, 217003 (2001).
  * [26] M. Zgirski, K.P. Riikonen, V. Touboltsev, and K.Y. Arutyunov, Phys. Rev. B 77, 054508 (2008).
  * [27] J.S. Lehtinen, T. Sajavaara, K. Yu. Arutyunov, M. Yu. Presnjakov, and A. L. Vasiliev, Phys. Rev. B 85, 094508 (2012).
  * [28] X.D.A. Baumans, D. Cerbu, O-A. Adami, V.S. Zharinov, N. Verellen, G. Papari, J.E. Scheerder, G. Zhang, V.V. Moshchalkov, A.V. Silhanek, J. Van de Vondel, Nat. Commun. 7, 10560 (2016).
  * [29] A. Radkevich, A.G. Semenov, and A.D. Zaikin, Phys. Rev. B 96, 085435 (2017).
  * [30] K.D. Usadel, Phys. Rev. Lett. 25, 507, (1970).
  * [31] W. Belzig, F. Wilhelm, C. Bruder, G. Schön, and A.D. Zaikin, Superlatt. Microstruct. 25, 1251 (1999).
  * [32] R.C. Dynes, V. Narayanamurti and J.P. Garno, Phys. Rev. Lett. 41, 1509 (1978).
  * [33] S.V. Panyukov and A.D. Zaikin, J. Low Temp. Phys. 73, 1 (1988).
  * [34] K.Yu. Arutyunov, J.S. Lehtinen, A.A. Radkevich, A.G. Semenov, and A.D. Zaikin, J. Magn. Magn. Mat. 459, 356 (2018).
  * [35] B. Camarota, F. Parage, F. Balestro, P. Delsing, and O. Buisson, Phys. Rev. Lett. 86, 480 (2001).
  * [36] A.G. Semenov and A.D. Zaikin, Phys. Rev. B 88, 054505 (2013).
  * [37] A.G. Semenov and A.D. Zaikin, J. Phys.: Condens. Matter 22, 485302 (2010).
  * [38] A.D. Zaikin and S.V. Panyukov, Phys. Lett. A 120, 306 (1987).
  * [39] D.V. Averin and A.A. Odintsov, Phys. Lett. A 140, 251 (1989).
  * [40] A.D. Zaikin, J. Low Temp. Phys. 80, 223 (1990).
  * [41] G. Schön and A.D. Zaikin, Phys. Rep. 198, 237 (1990).
  * [42] J.E. Mooij and Yu.V. Nazarov, Nat. Phys. 2, 169 (2006).
  * [43] A. G. Semenov and A. D. Zaikin, Low Temp. Phys. 43, 805 (2017).
  * [44] A. Barone and G. Paterno, Physics and Applications of the Josephson Effect (Wiley, New York, 1982).
  * [45] O.V. Astafiev, L.B. Ioffe, S. Kafanov, Yu.A. Pashkin, K.Yu. Arutyunov, D. Shahar, O. Cohen, and J.S. Tsai, Nature 484, 355 (2012).
  * [46] J.T. Peltonen, O.V. Astafiev, Yu.P. Korneeva, B.M. Voronov, A.A. Korneev, I.M. Charaev, A.V. Semenov, G.N. Golt’sman, L.B. Ioffe, T.M. Klapwijk, and J.S. Tsai, Phys. Rev. B 88, 220506(R) (2013).
  * [47] J.S. Lehtinen, K. Zakharov, and K.Yu. Arutyunov, Phys. Rev. Lett. 109, 187001 (2012).
  * [48] T.T. Hongisto and A.B. Zorin, Phys. Rev. Lett. 108, 097001 (2012).
  * [49] S.E. de Graaf, S.T. Skacel, T. Hönigl-Decrinis, R. Shaikhaidarov, H. Rotzinger, S. Linzen, M. Ziegler, U. Hübner, H.-G. Meyer, V. Antonov, E. Il’ichev, A.V. Ustinov, A.Ya. Tzalenchuk, and O.V. Astafiev Nat. Phys. 14, 590 (2018).
  * [50] Z. Wang, J.S. Lehtinen, and K.Yu. Arutyunov, Appl. Phys. Lett. 114, 242601 (2019).
  * [51] For this very reason the notion "coherent quantum phase slips" sometimes used in the literature is to a large extent tautological. No such objects as "incoherent quantum phase slips" (or, by duality, "incoherent Cooper pairs") exist in Nature. At the same time, quantum tunneling of both fluxons and Copper pairs can, of course, be made incoherent provided, e.g., enough extra dissipation is added to the system.
  * [52] A.G. Semenov and A.D. Zaikin, Phys. Scr. T151, 014022 (2012).
  * [53] J.E. Mooij and C.J.P.M. Harmans, New J. Phys. 7, 219 (2005).
  * [54] A.G. Semenov and A.D. Zaikin, Phys. Rev. B 84, 045416 (2011).
  * [55] D. Averin and H.T. Imam, Phys. Rev. Lett. 76, 3814 (1996).
  * [56] A. Martin-Rodero, A. Levy Yeyati, and F.J. Garcia-Vidal, Phys. Rev. B 53, R8891 (1996).
  * [57] A.V. Galaktionov and A.D. Zaikin, Phys. Rev. B 82, 184520 (2010).
  * [58] Ya. M. Blanter and M. Büttiker, Phys. Rep. 336, 1 (2000).
  * [59] A.G. Semenov and A.D. Zaikin, Phys. Rev. B 94, 014512 (2016).
  * [60] U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 2008), p. 207.
  * [61] D. Controzzi, F.H.L. Essler, and A.M. Tsvelik, Phys. Rev. Lett. 86, 680 (2001).
  * [62] A. G. Semenov and A. D. Zaikin, Fortschr. Phys. 65, 1600043 (2017).
  * [63] A. G. Semenov and A. D. Zaikin, J. Supercond. Nov. Magn. 30, 139 (2017).
  * [64] A. G. Semenov and A. D. Zaikin, J. Supercond. Nov. Magn. 31, 711 (2018).
  * [65] D.S. Golubev, M. Marthaler, Y. Utsumi, and G. Schön, Phys. Rev. B 81, 184516 (2010).
  * [66] M. Zonda, W. Belzig and T. Novotny, Phys. Rev. B 91, 134305 (2015).
  * [67] A.G. Semenov and A.D. Zaikin, Phys. Rev. B 99, 094516 (2019).
  * [68] A.V. Galaktionov, D.S. Golubev, and A.D. Zaikin, Phys. Rev. B 68, 235333 (2003).
  * [69] D.V. Averin, Yu.V. Nazarov, and A.A. Odintsov, Physica B 165-166, 945 (1990).
  * [70] P. A. Bobbert, R. Fazio, G. Schön, and A. D. Zaikin, Phys. Rev. B 45, 2294 (1992).
  * [71] A. Radkevich, A.G. Semenov, and A.D. Zaikin, Phys. Rev. B 100, 014520 (2019).
  * [72] H. Kleinert, EJTP 8, 57 (2011).
  * [73] M.P.A. Fisher and W. Zwerger, Phys. Rev. B 32, 6190 (1985).
  * [74] A. Schmid, Phys. Rev. Lett. 51, 1506 (1983).
  * [75] S.V. Panyukov and A.D. Zaikin, Physica B 152, 162 (1988).
  * [76] S. Lukyanov and A. Zamolodchikov, Nucl. Phys. B 493, 571 (1997).
  * [77] A.O. Gogolin, A.A. Nersesyan, and A.M. Tsvelik, Bosonization and Strongly Correlated Systems (Cambridge University Press, Cambridge, 1998).
  * [78] A. Radkevich, A.G. Semenov, and A.D. Zaikin, J. Supercond. Nov. Magn. 33, 2335 (2020).
  * [79] F.W.J. Hekking and L.I. Glazman, Phys. Rev. B 55, 6551 (1997).
  * [80] F. Guinea and G. Schön, EPL 1, 585 (1986).
  * [81] S.V. Panyukov and A.D. Zaikin, Phys. Rev. Lett. 67, 3168 (1991).
  * [82] W. Hofstetter and W. Zwerger, Phys. Rev. Lett. 78, 3737 (1997).
  * [83] M. Kindermann and Yu.V. Nazarov, Phys. Rev. Lett. 91, 136802 (2003).
  * [84] D.S. Golubev and A.D. Zaikin, Phys. Rev. B 69, 075318 (2004).
  * [85] D.A. Bagrets and Yu.V. Nazarov, Phys. Rev. Lett. 94, 056801 (2005).
  * [86] S.A. Bulgadaev, JETP Lett. 39, 315 (1984).
  * [87] F. Guinea, V. Hakim, and A. Muramatsu, Phys. Rev. Lett. 54, 263 (1985).
  * [88] A. Latyshev, A.G. Semenov, and A.D. Zaikin, J. Supercond. Nov. Magn. 33, 2329 (2020).
  * [89] A. Latyshev, A.G. Semenov, and A.D. Zaikin, Beilstein J. Nanotech. 11, 1402 (2020).
  * [90] A. Latyshev, A.G. Semenov, and A.D. Zaikin, Beilstein J. Nanotech. 13, 292 (2022).
  * [91] C. Delacour, B. Pannetier, J.-C. Villegier, and V. Bouchiat, Nano Lett. 12, 3501 (2012).
  * [92] K.Yu. Arutyunov, J.S. Lehtinen, A.A. Radkevich, A.G. Semenov, and A.D. Zaikin, Commun. Phys. 4, 146 (2021).
  * [93] A.T. Bollinger, R.C. Dinsmore, A. Rogachev, and A. Bezryadin, Phys. Rev. Lett. 101, 227003 (2008).