Nonequilibrium Charge Noise and Dephasing from a Spin-Incoherent Luttinger Liquid

PHYSICAL REVIEW B 75, 035336 ͑2007͒

Nonequilibrium charge noise and dephasing from a spin-incoherent Luttinger liquid

Gregory A. Fiete1,2 and Markus Kindermann3,4 1Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA 2Department of Physics, California Institute of Technology, MC 114-36, Pasadena, California 91125, USA 3Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853-2501, USA 4School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA ͑Received 29 September 2006; revised manuscript received 30 November 2006; published 29 January 2007͒ We theoretically investigate the charge noise and dephasing in a metallic device in close proximity to a spin incoherent Luttinger liquid with a small but ﬁnite current. The frequency dependence of the charge noise exhibits a loss of frequency peaks corresponding to the 2kF part of the density correlations in the electron liquid when the temperature T is increased from values below the magnetic exchange energy J of the electron gas to values above it. The dephasing rate in a nearby metallic nanostructure also shows a crossover for T ϳJ and may exhibit a nonmonotonic temperature dependence. For a range of temperatures the dephasing rate decreases with increasing temperatures. The proposed experiments provide a convenient approach to probe the spin-incoherent Luttinger liquid and should be implementable in a wide variety of systems.

DOI: 10.1103/PhysRevB.75.035336 PACS number͑s͒: 73.21.Ϫb, 71.27.ϩa, 71.10.Pm

I. INTRODUCTION rial. At large rs there can be an exponentially large separation of spin and charge energy scales.14,15 At finite temperatures, The noise of the electrical current has proven to be a this makes it possible to have highly excited spin degrees of powerful probe of small electronic structures.1,2 While the freedom while keeping the charge degrees of freedom close equilibrium or Johnson-Nyquist noise only contains informa- to the ground state. In 1D this scenario is referred to as the tion about the temperature and the linear response of a sys- “spin incoherent Luttinger liquid” ͑SILL͒.16 It may also be tem, the fluctuations out of equilibrium ͑such as in the pres- realized at high electron densities in very thin wires.17 ence of a voltage bias͒ carries a great wealth of information. The SILL has received much attention recently because of For example, the nonequilibrium current ͑shot͒ noise of a its distinct properties that partially resemble those of a Lut- conductor reveals the granularity of electric charge and can be used to measure its fundamental value. Such experiments have proved extremely useful in demonstrating convincingly that the ␯=1/͑2n+1͒ fractional quantum Hall effect states have a basic unit of charge e/͑2n+1͒.3–5 Measurements of the noise can also be used to determine the particle statistics of many-body systems with effects such as “bunching” for bosons6 and “antibunching” for fermions7,8 predicted and observed. Finite-frequency noise measurements as discussed in this article have been proposed before to observe fractional charges in nonchiral Luttinger liquids9,10 as well as many- body resonances in interacting nanostructures.11 In this paper we study a schematic situation like that shown in Figs. 1͑a͒ and 1͑b͒. A current I is driven in a one dimensional ͑1D͒ system such as a carbon nanotube or semiconductor quantum wire, and a small device such as a me- FIG. 1. Schematic of the proposed experiments described in the tallic gate or a charge qubit is placed in close proximity.12,13 text and the dimensionless potential u͑x͒ at the gate/qubit due to a Due to the discrete nature of the electrons ͑quasiparticles͒ in charge density ␦␳͑x͒ at position x along the 1D system. A current I the 1D system, there will be a time-dependent charge Q͑t͒ is assumed to flow in the 1D system as indicated. This could be a induced on the gate in Fig. 1͑a͒ and a time-dependent poten- carbon nanotube or semiconductor quantum wire, for example. In ͑ ͒ tial that dephases the charge states of the qubit in Fig. 1͑b͒. a a metallic gate is placed nearby and the fluctuations in the in- Q͑t͒ CV ͑t͒ e͐dxu͑x͒␦␳͑x t͒ e We will address the frequency spectrum of the charge fluc- duced charge = g = , are measured. Here is the charge of the electron and C is the capacitance. In ͑b͒ a charge tuations ͗Q͑t͒Q͑0͒͘ on the gate in Fig. 1͑a͒ and the decoher- ͑ ͒ qubit is prepared in a quantum coherent state and its decoherence ence rate of the qubit in Fig. 1 b . time is measured. In ͑c͒ u͑x͒ and its Fourier transform ͑d͒ are We are interested in the particular situation where the in- shown. In order to observe the effects described in this paper, the ͑ teractions between electrons in the wire we will refer to the size of the gate and qubit, and the distance to the 1D system should 1D system generically in this paper as a wire͒ are very strong be small compared to the interparticle spacing in the 1D system a. and appreciable Wigner-crystal-like correlations are present. This ensures the Fourier components of the potential u͑2k ͒ and ͑ ͒ F This typically means that rs =a/ 2aB is large, where a is the u͑4kF͒ and their corresponding contributions to noise/dephasing are interparticle spacing and aB is the Bohr radius for the mate- not too small.

1098-0121/2007/75͑3͒/035336͑9͒ 035336-1 ©2007 The American Physical Society GREGORY A. FIETE AND MARKUS KINDERMANN PHYSICAL REVIEW B 75, 035336 ͑2007͒ tinger liquid ͑LL͒ but partially do not.18–29 The crucial dif- dx 1 ͒ ͑ ␾ ͑ ͔͒2ͬ ץ͓ ␪ ͑ ͔͒2 ץ͓ ͫ ͵ eff ប 32–30 ference between the more familiar LL and a SILL is the Hs = vs x s x + Ks x s x , 4 2␲ K combination of ͑i͒ very strong particle-particle interaction s ͑ ͒ Ϸ ប ͑ ͒ and ii finite but small temperature. As stated above, in a where vs Ja/ , Ks =1 for SU 2 symmetry, and the bosonic SILL the charge degrees of freedom are only very weakly spin fields satisfy the same commutation relations as the influenced by the temperature, while the spin degrees of free- charge fields and they commute with the charge fields. In Eq. ͑ ͒ ␻ dom are highly thermally excited. In fact, below a critical 3 the characteristic frequency of lattice oscillations 0 is ␻ 23 temperature related to the fundamental energy scale in the related to the charge velocity as vc = 0a. Note that in the spin sector, LL behavior is obtained in the same system ex- effective description of the 2kF part of the density oscilla- hibiting SILL behavior at higher temperatures. This fact en- tions ͑3͒, the amplitude is suppressed by a factor ables us to study the SILL within an effective low energy LL ϳ͑ J1 ͒ 2 2 1 relative to the corresponding value one would m␻0a description as it is “approached” by increasing the tempera- obtain from bosonizing a weakly interacting electron gas.30 ture from below the critical value. The details of this method Returning to the main task of this paper, computing the of study are described below. They have already proven use- 22 charge fluctuation spectrum on the gate and the decoherence ful in illuminating the electrical transport and Coulomb rate of the qubit,33 we will find that both are determined by drag23 properties in SILL systems. ͑ the Fourier transform of the density-density correlation func- The generic form of the single mode nonchiral right and tion ͗͑␳eff͑x,t͒−␳ ͒͑␳eff͑0,0͒−␳ ͒͘. The key physical point is left movers present͒ 1D Hamiltonian for energies small com- 0 0 that when JTE , the 2k parts of this correlation func- pared to the charge scale, but arbitrary compared to the spin c F tion will be thermally washed out and the effects of losing scale is H =H +H , where elec c s these correlations will be observed in the charge fluctuation ͑noise͒ spectrum on the gate and the decoherence rate of the dx 1 ␾ ͑x͔͒2ͮ ͑1͒ qubit. In particular we find that the sharp gate response at the ץ͓ ␪ ͑x͒͒2 + K ץ͑ ͭ ͵ H = ប v c c 2␲ K x c c x c c frequencies corresponding to the 2kF oscillations vanishes, and the decoherence time of the qubit may exhibit nonmono- and tonic behavior, including a region where the decoherence time ͑rate͒ increases ͑decreases͒ as the temperature increases, ជ ជ ͑ ͒ Hs = ͚ JlSl · Sl+1. 2 counter to the naive expectation. See Fig. 3. l This paper is organized as follows. In Sec. II we introduce the basic formalism for calculating the nonequilibrium noise ͑ ͒ The Hamiltonian 1 describes the low-energy density fluc- in a situation like that in Fig. 1. In Sec. III we discuss the tuations of the electron gas. The energy scale for the charge behavior of the noise for strongly interacting electrons. As ϳ ប ប sector is set by Ec vc /a, where is Planck’s constant, vc the SILL state is approached from temperatures TϽJ we is the collective charge mode velocity, and a is the average describe the resulting crossover in the noise spectrum when spacing between electrons. The parameter Kc describes the TϳJ. In Sec. IV we describe the formalism for computing strength of the microscopic interactions and the bosonic the dephasing ͑decoherence͒ time ␶␸ for a charge qubit in fields appearing in Eq. ͑1͒ satisfy the commutation relations close proximity to a current carrying wire and evaluate ␶␸ as ␪ ͑ ͒ ␾ ͑ Ј͔͒ ␲␦͑ Ј͒ ץ͓ x c x , c x =i x−x . For strong interactions the spin a function of temperature. Finally, in Sec. V we describe the degrees of freedom behave as a 1D antiferromagnetic spin main conclusions of the work and prospects for experimental ͑ ͒ ជ chain. In Eq. 2 , Sl is the spin of the lth electron, and Jl is implementation. the nearest neighbor exchange energy which depends on the local separation of electrons. As discussed in Ref. 23,to lowest order in local electron displacement from equilibrium II. DESCRIBING THE FLUCTUATIONS J ϷJ+J ͑u −u ͒, where u is the displacement of the lth l 1 l+1 l l In real physical systems the charge density is not uniform, electron. The main role of the term proportional to J is to 1 so when current flows along a 1D system as shown in Fig. 1 induce 2k oscillations in the density correlations after the F time-dependent fluctuations in the potential outside the sys- higher energy density fluctuations of the electron gas have tem are created. If a metallic gate is nearby ͓Fig. 1͑a͔͒, these been integrated out.23 At the lowest energies TϽJ,E , the c fluctuations will cause a time-dependent induced charge Q͑t͒ effective density is given by23 on it.12,13 If a charge qubit is nearby ͓Fig. 1͑b͔͒ these same fluctuations will lead to decoherence or dephasing. Our main ͱ2 ͒ ͑ ␪ ץ ␳eff͑ ͒ ␳ x,t = 0 − x c x,t goal in this section is to lay out the relevant formalism for ␲ describing the nonequilibrium fluctuations on the gate and 34 J the qubit. − ␳ ͩ 1 ͪsin͓2k x + ͱ2␪ ͑x,t͔͒sin͓ͱ2␪ ͑x,t͔͒ 0 ␻2 2 F c s Since the gate is assumed metallic, a net local charge m 0a imbalance in the wire, e␦␳͑x,t͒dx, will induce a charge ␳ ͓ ͱ ␪ ͑ ͔͒ ͑ ͒ ␦ ͑ ͒ + 0cos 4kFx + 8 c x,t . 3 Q t on the gate whose magnitude will depend on the proximity of the local charge in the wire to the gate. We call the At low energies TϽJ the effective Hamiltonian of the SU͑2͒ function describing the distance dependence u͑x͓͒see Fig. spin sector ͑2͒ is23 1͑c͒ for a schematic͔ and it has the same dependence on x as

035336-2 NONEQUILIBRIUM CHARGE NOISE AND DEPHASING… PHYSICAL REVIEW B 75, 035336 ͑2007͒ the interaction potential between the charge e␦␳͑x,t͒dx and ␦Q͑t͒. The total charge on the gate is given by summing all the contributions along the wire

Q͑t͒ = e ͵ dxu͑x͒␦␳͑x,t͒. ͑5͒

The charge fluctuations ͑noise͒ on the gate are thus determined by ͕͗Q͑t͒,Q͑0͖͒͘=e2 ͐dx͐dxЈu͑x͒u͑xЈ͒ ϫ͕͗␦␳͑x,t͒,␦␳͑xЈ,0͖͒͘, where ͕A,B͖=AB+BA and the ex- FIG. 2. Schematic of a snapshot of the charge density ␦␳͑x,t͒ in a strongly interacting 1D system flowing by a metallic gate with pectation values are to be evaluated in the presence of a finite Q͑t͒ ͑ ͒ e ͐ ͑ ͒␦␳͑ ͒͑ ͒ current I. The frequency-dependent charge noise induced voltage Vg t = C = C dxu x x,t see Fig. 1 . The current Iϵevd /a in the 1D system produces strong voltage and charge 1 ␻ ͑ ␲ ͒ ␲ ͑␻͒ i␻t͕͗ ͑ ͒ ͑ ͖͒͘ ͑ ͒ spikes on the gate with frequency I =vd 2 /a =2 I/e, corre- SQ = ͵ dte Q t ,Q 0 6 ϵ ␲ 2 sponding to 4kF 2 /a density modulations. Here vd is the mean velocity of the charge e due to the current I. can be expressed as e2 ␹͑q,␻͒ + ␹͑q,− ␻͒ washed out and this will cause the vanishing of the ␻ /2 ͑␻͒ ͵ ͉ ͑ ͉͒2 ͑ ͒ I SQ = dq u q , 7 2␲ 2 contribution to the noise and a loss of the 2kF contribution to ␶␸. We now turn to a quantitative discussion of these points. where ␹͑q,␻͒ is the Fourier transform of the density-density correlation function ␹͑x,t͒=͗␦␳͑x,t͒␦␳͑0,0͒͘. Here ␦␳͑x,t͒ ␳͑ ͒ ␳ ␳ ϵ III. NOISE SPECTRUM = x,t − 0, with 0 1/a, and a is the interparticle spacing. ͕͗ ͑ ͒ ͑ ͖͒͘ Note that since Q t ,Q 0 is manifestly real and it only As we have seen in the previous section, the noise S ͑␻͒ ͉ ͉ ͑␻͒ ͉͑␻͉͒ Q depends on t , SQ =SQ is real. is determined by the equilibrium correlation function ␹ ͑ ␻͒ When a current I is flowing through the wire, the elec- 0 q, , and this has dominant low frequency parts coming trons are moving with a mean velocity v =aI/e. This veloc- Ϸ d from wave vectors near q 0, 2kF, and 4kF. In this section ity sets a characteristic frequency scale ͑discussed below͒ ␹ ͑ ␻͒ we compute the various components of 0 q, at finite tem- which will appear in the noise spectrum. The density-density perature and use them to extract the frequency dependence of ␹͑ ␻͒ ͑ ͒ correlation function q, appearing in Eq. 7 can be ob- S ͑␻͒. We will focus on the case where the interactions are ␹ ͑ ␻͒ Q tained from the equilibrium correlation function 0 q, by strong and the density fluctuations are described by Eq. ͑3͒. 12 making a Galilean transformation so that The noise in the case of weak interactions and the case of ␹͑ ͒ ␹ ͑ ͒͑͒ strong interactions without the spin are discussed in Ref. 12, x,t = 0 x − vdt,t 8 both at zero temperature. Recent advances in computing the or, equivalently, density-density correlation function for weakly interacting 35,36 37 ␹͑q,␻͒ = ␹ ͑q,␻ − v q͒, ͑9͒ systems and for special forms of the interaction may be 0 d used in Eq. ͑7͒ to compute the noise in these cases. which can then be substituted into Eq. ͑7͒. The problem of computing the nonequilibrium noise on the gate or the qubit ␹qÉ0 ␻ ␻ is thus reduced to the problem of finding the equilibrium A. The 0 „q, … contribution to SQ„ … ␹ ͑ ␻͒ correlation function 0 q, . Focusing first on the qϷ0 components in Eq. ͑3͒,we Based on the structure of the expression for the effective compute the Fourier transform of ␹qϷ0͑x,t͒ ͱ ͱ 0 ␪ ͑ ͒͘ ץ ␪ ͑ ͒ 2 ץ density, ͑3͒, we expect the Fourier transform of the density- ͗ 2 = ␲ x c x,t ␲ x c 0,0 . Using the Fourier expansion density correlation function in the strong interaction limit ͑ ͒ appropriate for realizing the SILL to have the form ␲ ប ␻ ␻ ␪ ͑ ͒ ͱ iqx͑ −i qt † i qt͒ Ϸ c x,t = ͚ e aqe + a−qe , ␹ ͑ ␻͒Ϸ␹q 0͑ ␻͒ ␹2kF͑ ␻͒ ␹4kF͑ ␻͒ ͑ ͒ ͱ 2aLm␻ 0 q, 0 q, + 0 q, + 0 q, , 10 2 q q plus higher order terms that are subdominant. Each of these ͑11͒ terms will lead to a characteristic frequency response in where L is the length of the 1D system, and ␻ =v ͉q͉ is the ͑␻͒ q c SQ that can already be anticipated on purely physical dispersion of the long wavelength density fluctuations. The grounds. In the strong interaction Wigner crystal limit, the bosonic operators satisfy the commutation relations dominant periodicity of the density fluctuations are the 4kF ͓a ,a† ͔=␦ . Computing the Fourier transform of ϵ ␲ q qЈ q,qЈ 2 /a pieces. Thus, for a given current I=evd /a there will ␹qϷ0͑ ͒ ␻ 0 x,t immediately leads to be an electron passing by the gate with frequency I ϵ ͑ ␲ ͒ ␲ ͑ ͒ 2 vd 2 /a =2 I/e see Fig. 2 . The 2kF oscillations will Ϸ q ␲ ប L ͓␦͑␻ − ␻ ͒ − ␦͑␻ + ␻ ͔͒ ␻ Ϸ ␹q 0͑q,␻͒ = q q , ͑12͒ pass by with 1/2 the frequency, I /2, and the q 0 fluctua- 0 a2 m␻ 1−e−␤␻ tions will be peaked about zero frequency. The central phys- q ͑ ͒ ͑ ͒ ics of our main results can now be seen: When T J the 2kF which can then be substituted into Eq. 7 using Eq. 9 to parts of the density-density correlation function will be yield the noise contribution

035336-3 GREGORY A. FIETE AND MARKUS KINDERMANN PHYSICAL REVIEW B 75, 035336 ͑2007͒

2 ͑ ͒ ␻ velocity vc. Expanding the result 13 in vd /vc yields current- ͯuͩ ͪͯ ͑ ͒2 e2 ␲ ប L͉␻͉ v − v dependent fluctuations only at order vd /vc . These can be qϷ0͑␻͒ ΄ c d ΅ qϷ0͑␻͉͒ qϷ0͑␻͉͒ SQ = 2 2 extracted by measuring SQ I −SQ I=0 2␲ 2a mv ͑v − v ͒ 2 c c d Ϸ e ␲បL͉␻͉ ͉ ͑ ␻ ͉͒2 ͉␻͉ ͑ vd ͒2 ␲ 2 u 6 . 2 2a mvc vc vc 1 1 ϫ 2k ͩ v − v ͪ΄ ␹ F ␻ ␻ d d B. The q, contribution to SQ −ͩ1+ ͪ␤͉␻͉ ͩ1+ ͪ␤͉␻͉ 0 „ … „ … 1−e v −v 1−e v −v c d c d ␹2kF͑ ␻͒ Ͻ The behavior of 0 q, for temperatures T J and T ͯ ␻ ͯ2 ϾJ is the central contributing factor to the interesting tem- uͩ ͪ ͑␻͒ ␶ vc + vd 1 perature dependence of SQ and ␸ when the 1D system + 2 ͩ v has interactions strong enough to produce a large separation ͑v + v ͒ ͩ d ͪ␤͉␻͉ c d − 1− 1−e vc+vd in spin and charge velocities. When vs vc and the system is at finite temperature, the spin-incoherent regime can be reached. As we have emphasized in the introduction, the 1 տ ͑ ͒ main effect in the density correlations at T J is that the 2kF − v ͪ΅. 13 d 23 ͩ1− ͪ␤͉␻͉ components get washed out by thermal effects and this 1−e v +v c d effectively eliminates these contributions to the nonequilib- qϷ0͑␻͒ϳ͉␻͉ ␻→ ␶ It is worth noting that SQ as 0 for realistic rium noise and ␸. Let us first see how the 2kF correlations forms of u͑x͒ which have finite u͑q=0͒. are lost before we turn to the T=0 correlations and the cor- In the limit of a small current I in the 1D system ͑to responding noise. ͒ ͑ ͒ ␹2kF͑ ␻͒ Ͻ Ͻ prevent the system from being too far out of equilibrium we From Eq. 3 , one sees that 0 q, for 0 T J Ϸ ប expect the velocity vd to be much smaller than the charge vs /a is given by the Fourier transform of

J 2 ͑␲Tr /v ͒Kc ͗␳eff ͑x,t͒␳eff ͑0,0͒͘ = ␳2ͩ 1 ͪ cos͑2k x͒ c c 2kF 2kF 0 2 2 F m␻ a Kc/2 0 ␲T ␲T ͭ ͫ ͑ ͒ͬ ͫ ͑ ͒ͬͮ sinh x − vct + irc sinh x + vct − irc vc vc ͑␲Tr /v ͒Ks ϫ s s , ͑14͒ K /2 ␲T ␲T s ͭ ͫ ͑ ͒ͬ ͫ ͑ ͒ͬͮ sinh x − vst + irs sinh x + vst − irs vs vs

͑ ͒Ͼ տ where the inﬁnitesimals rc ,rs =O a 0. clearly that when T J, the 2kF density correlations are ex- In order to see how temperature affects the 2kF correla- ponentially suppressed. When vs vc, one can take the zero tions, it is instructive to consider the equal time correlation temperature limit in the charge sector20 ͑since one can simul- ͉ ͉ Ͼ ␲ ͉ ͉ ␲ ͉ ͉ ͒ functions for x a,rc ,rs, taneously have T x /vs 1 and T x /vc 1 and only an exponentially small error is made. This approximation J 2 ͗␳eff ͑ ͒␳eff ͑ ͒͘ Ϸ ␳2ͩ 1 ͪ ͑ ͒ gives x,0 0,0 cos 2kFx 2kF 2kF 0 m␻2a2 0 Ks ͑ ͒ T cos 2kFx K K ͗␳eff ͑ ͒␳eff ͑ ͒͘ Շ ͩ ͪ −cT/J ͑ ͒ ͑␲ ͒ c ͑␲ ͒ s x,0 0,0 e . 16 Trc/vc Trs/vs 2kF 2kF ͑ ͒Kc ϫ , J x/rc ␲T Kc ␲T Ks ͫsinhͩ xͪͬ ͫsinhͩ xͪͬ Equation ͑15͒ is not applicable to temperatures of the order vc vs of J or larger because the bosonized description of the spin 2 ␲ Ks ␲ Kc J1 Trs Trc excitations ceases to be valid. One can infer, however, from Ͻ ␳2ͩ ͪ ͩ ͪ ͩ ͪ ͑ ͒ 0 m␻2a2 v v the solution of the original spin Hamiltonian 2 , that the 0 s c exponential decay continues at TϾJ.38 cos͑2k x͒ This exponential decay with temperature carries over to ϫ e−cT/J F , ͑15͒ ␲T Kc the Fourier transform of the density-density correlation func- ͫsinhͩ xͪͬ tion at wave vector 2k , ␹2kF͑q,␻͒. We turn now to the zero v F 0 c ␹2kF temperature limit of 0 . To do so, we must compute the where cϾ0 is a constant of order unity. Equation ͑15͒ shows Fourier transform

035336-4 NONEQUILIBRIUM CHARGE NOISE AND DEPHASING… PHYSICAL REVIEW B 75, 035336 ͑2007͒

2 ϱ ϱ Kc Ks J1 ͑␻ ͒ rc rs ␹2kF͑ ␻͒Ϸ␳2ͩ ͪ ͵ ͵ i t−q±x ͑ ͒ ¯ q±, dx dte , 17 0 0 ␻2 2 ͓͑ ͒͑ ͔͒Kc/2 ͓͑ ͒͑ ͔͒Ks/2 m 0a −ϱ −ϱ x − vct + irc x + vct − irc x − vst + irs x + vst − irs

␹2kF͑ ␻͒ ␹2kF͑ ␻͒ ␹2kF͑ ␻͒ where q± =q±2kF, and 0 q, =¯ 0 q+ , +¯ 0 q− , . This is shown in schematic form in Fig. 8 of the review ͑ ͒ 30 From Eq. 17 one can see that the singularity near x=vct article by Voit, but there appears to be a typo in the expo- ϳ͉␻ ͉Kc/2+Ks−1 ␻Ϸ yields a contribution −vcq± and the singularity nent for ±vcq±. ϳ͉␻ ͉Kc+Ks/2−1 ␹2kF͑ ␻͒ near x=vst yields a contribution −vsq± . The sin- The zero temperature analytical properties of ¯ 0 q± , gularities at x=−vct,−vst have the same behavior only with can be extracted by making the change of variables z=x → ͑ ͒ q± −q±. For SU 2 symmetry Ks =1, and one sees the con- +v t and ¯z=x−v t and deﬁning ␻Ϸ s s tribution from the spin singularity ±vsq± leads to a diver- gence in ¯␹2kF͑q,␻͒ only for K Ͻ1/2. Otherwise, it has an ␩ ϵ Ͻ ͑ ͒ 0 c vs/vc 1, 18 inverted cusp form. On the other hand, the contribution from ␻Ϸ the charge singularity, ±vcq±, always has a cusp form. which gives

ϱ ␻ ϱ ͑␻ ͒ K K ␳2 J 2 e−i ¯z/vs ei −vsq± x/vsr s r c ␹2kF͑ ␻͒Ϸ 0 ͩ 1 ͪ ͵ ͵ s c ¯ 0 q±, dz¯ dx . ␻2 2 ͑ ͒Ks/2 ͑ ͒Ks/2 ͓͑ ͑ ␩−1͒ ␩−1 ͒͑ ͑ ␩−1͒ ␩−1 ͔͒Kc/2 2vs m 0a −ϱ ¯z + irs −ϱ 2x −¯z − irs x 1− + ¯z + irc x 1+ − ¯z − irc ͑19͒ Note that when the outer integral over ¯z is taken all the singularities lie in the lower half-plane and therefore the result is proportional to ␪͑␻͒. If we further eliminate x in favor of ¯z and z,weﬁnd

ϱ ͑␻ ͒ ͑ ͒ K ϱ ͑␻ ͒ ͑ ͒ K ␳2 J 2 e−i +vsq± ¯z/ 2vs r s/2 ei −vsq± z/ 2vs r s/2 ␹2kF͑ ␻͒Ϸ 0 ͩ 1 ͪ ͵ s ͵ s ¯ 0 q±, dz¯ dz ␻2 2 ͑ ͒Ks/2 ͑ ͒Ks/2 2vs m 0a −ϱ ¯z + irs −ϱ z − irs rKc ϫ c . ͑20͒ ͕͓͑ ␩−1͒ ͑ ␩−1͒ ͔͓͑ ␩−1͒ ͑ ␩−1͒ ͔͖Kc/2 1− z + 1+ ¯z + i2rc 1+ z + 1− ¯z − i2rc Since ␩−1Ͼ1, when the z integral is taken, all the singularities lie in the upper half-plane giving a result proportional to ␪͑␻ ͒ −vsq± and when the ¯z integral is taken all the singularities lie in the lower have plane giving a result proportional to ␪͑␻ ͒ ␹2kF͑ ␻͒ϰ␪͑␻͒␪͓␻2 ͑ ͒2͔ Ͻ +vsq± . Combining these results, we ﬁnd ¯ 0 q± , − vsq± . For repulsive interactions, that is for Kc 1, Ϸ ͑ ͒ ␣ ␣Ͻ Ks 1, all singularities in Eq. 20 have the form 1/z with 1. The integrals thus converge and after having analysed the ͑ ͒ ␪͑␻͒␪͓␻2 ͑ ͒2͔ → analytic properties of Eq. 20 , resulting in the factor − vsq± , the limit rc ,rs 0 may be taken,

␹2kF͑ ␻͒ ϰ ␪͑␻͒␪͓␻2 ¯ 0 q±, ϱ ͑␻ ͒ ͑ ͒ ϱ ͑␻ ͒ ͑ ͒ e−i +vsq± ¯z/ 2vs ei −vsq± z/ 2vs 1 ͑ ͒2͔͵ ͵ − vsq± dz¯ dz . Ks/2 Ks/2 −1 −1 −1 −1 Kc/2 −ϱ ¯z −ϱ z ͕͓͑1−␩ ͒z + ͑1+␩ ͒¯z͔͓͑1+␩ ͒z + ͑1−␩ ͒¯z͔͖ ͑21͒

2k ͑1−͒ To obtain S we first evaluate S F ͑␻͒, the q contribution ␻ Q Q − Ϸ ␻ I/2 Ϸ ␻ ␹2kF ͑ ͒ the ratio vd /vs. Thus, qmin − v +2kF + v and qmax v to in the first term of Eq. 7 , ␻ s s s I/2 ␻ ϵ ͑ ␲ ͒ +2kF − v , where I vd 2 /a was defined at the end of 2 s ͑ ͒ e 1 Sec. II. Shifting q by 2k then gives 2kF 1− ͑␻͒ ͵ ͉ ͑ ͉͒2␹2kF͑ ␻ ͒ F SQ = dq u q ¯ 0 q −2kF, − vdq . 2␲ 2 ␻ ␻ 2 /vs− I/2/vs ͑ ͒ e 1 ͑ ͒ 2kF 1− ͑␻͒ ͵ 22 SQ = dq 2␲ 2 ͑␻ ␻ ͒ − /vs− I/2/vs The theta functions in Eq. ͑21͒ constrain the q integration to 2 2k ␻ ␻ ϫ ͉u͑q +2k ͉͒ ¯␹ F͑q,␻ − ␻ /2 − v q͒. ͑23͒ Ͻ Ͻ −vs2kF +vs2kF F 0 I d qmin q qmax where qmin= v −v and qmax= v +v . For Ͻ d s s c ␻ small currents, we expect vd vs, so that we can expand in We now extract the asymptotic behavior of SQ near

035336-5 GREGORY A. FIETE AND MARKUS KINDERMANN PHYSICAL REVIEW B 75, 035336 ͑2007͒

Ϸ Ϸ␻ /2. For this we assume that ͑␻−␻ ͒/2v 2k .Ifu͑q͒ ͑␻͒ q 0͑␻͒ 2kF͑␻͒ 4kF͑␻͒͑͒ I I s F SQ = SQ + SQ + SQ 30 is analytic at q=2kF, which we assume, we may for this ͑ ͒ ͑ ͒ ͑ ͒ with replace u q by u 2kF in Eq. 23 . By scaling the integration → ͑␻ ␻ ͒ → ͑␻ ␻ ͒ ͑␻ Ϸ variables by q q/ − I /2 and z,¯z z/ − I /2 ,¯z/ Sq 0͑␻͒ ϰ ͉u͑0͉͒2͉␻͉, ␻ ͒ ͑ ͒ Q − I /2 in 21 the asymptotic scaling of the resulting integral may then be extracted: J 2 ␻ Ks+Kc−1 2kF͑␻͒ ϰ ͩ 1 ͪ ͉ ͑ ͉͒2ͯ␻ I ͯ 2 SQ 2 2 ͚ u 2kF ± , ͑ ͒ J1 m␻ a 2 S2kF 1− ͑␻͒ϳͩ ͪ ͉u͑2k ͉͒2͉␻ − ␻ /2͉Ks+Kc−1. 0 ± Q ␻2 2 F I m 0a ͑ ͒ 4kF͑␻͒ ϰ ͉ ͑ ͉͒2͉␻ ␻ ͉4Kc−1 ͑ ͒ 24 SQ ͚ u 4kF ± I . 31 ± ␹2kF͑ Carrying through the same calculation for the ¯ 0 q ␻ ͒ The frequency dependence of the charge noise measured at a +2kF , −vdq contribution one finds gate nearby a quantum wire thus displays power law singu- 2 ͑ ͒ J1 larities at the frequencies ␻ /2 and ␻ that are observable at S2kF 1+ ͑␻͒ϳͩ ͪ ͉u͑2k ͉͒2͉␻ + ␻ /2͉Ks+Kc−1. I I Q ␻2 2 F I low temperatures, kT␻,␻ . The singularity S2kF at ␻ m 0a I Q Ϸ␻ /2, however, becomes exponentially small as TտJ. ͑25͒ I Likewise, the contributions from the second term in Eq. ͑7͒ IV. DEPHASING take on a similar form and we finally arrive at the zero tem- Having discussed the finite frequency noise in a metallic perature result gate such as the one in Fig. 1͑a͒, we now turn to the situation J 2 ␻ Ks+Kc−1 shown in Fig. 1͑b͒. As we will see immediately below, the S2kF͑␻͒ϳͩ 1 ͪ ͚ ͉u͑2k ͉͒2ͯ␻ ± I ͯ . ␶ Q m␻2a2 F 2 dephasing rate ␸ of the charge qubit depicted there is deter- 0 ± mined by the zero frequency noise produced by the nearby ͑26͒ quantum wire.39,40 We assume an effective coupling Hamil- ͑ ͒ tonian between wire and qubit of the form Equation 26 shows that indeed the 2kF contributions to the ␻ noise are centered about ± I /2 with a power-law behavior e2 ␴ ͩ ͪ͵ ͑ ͒␦␳͑ ͒ ͑ ͒ that depends on the interaction parameters of the 1D system. Hwq = z dxu x x , 32 2C ␴ where z is the third Pauli matrix acting on the space of qubit C. The ␹4kF q,␻ contribution to S ␻ 0 „ … Q„ … states. The dephasing rate ␶␸ of the qubit is then determined The next dominant component of the noise comes from by the zero frequency fluctuations of ␦␳, the 4k part of the density correlations ͑3͒, ϱ F 1 1 = ͵ dtK͑t͒, ͑33͒ ͗␳eff ͑x,t͒␳eff ͑0,0͒͘ ␶ 4kF 4kF ␸ 2 −ϱ ␳2 ͑ ͒ = 0cos 4kFx where ͑␲ ␣ ͒4Kc T /vc e2 2 ϫ . ͑ ͒ ͩ ͪ ͵ Ј ͑ ͒ ͑ Ј͒͗␦␳͑ ͒␦␳͑ Ј ͒͘ 2K K t = dxdx u x u x x,t x ,0 , ␲T ␲T c 2C ͭ ͫ ͑ ͒ͬ ͫ ͑ ͒ͬͮ sinh x − vct + irc sinh x + vct − irc ͑ ͒ vc vc 34 ͑ ͒ 27 so that Taking the zero temperature limit of this correlation function 1 1 e2 2 dq and evaluating the Fourier transform after making the coor- = ͩ ͪ ͵ ͉u͑q͉͒2␹͑q,␻ =0͒ ␶␸ 2 2C 2␲ dinate transformation z=x+vct, ¯z=x−vct one finds 2 2 ␹4kF͑ ␻͒Ϸ␪͑␻͒␪͓␻2 ͑ ͒2͔͓␻2 ͑ ͒2͔2Kc−1 1 e dq q±, − vcq± − vcq± ͩ ͪ ͉ ͑ ͉͒2␹ ͑ ͒ ͑ ͒ 0 = ͵ u q q,− v q . 35 2 2C 2␲ 0 d ␳2 0 4K 2 2 Ϸ ϫ r c⌫͑1−2K ͒ sin͑2␲K ͒ , ͑28͒ ␹ ͑ ␻͒Ϸ␹q 0͑ ␻͒ ␹2kF͑ ␻͒ c c c As we have seen earlier, 0 q, 0 q, + 0 q, vc ␹4kF͑ ␻͒ + 0 q, , plus higher order terms that are subdominant. In ␹4kF͑ ␻͒ ␹4kF͑ ␻͒ ␹4kF͑ ␻͒ where again we have 0 q, =¯ 0 q+ , +¯ 0 q− , . the expression for the dephasing rate 1/␶␸ each of these After the Galilean shift ͑9͒ we find terms is multiplied by the appropriate factor of ͉u͑q͉͒2.As shown in Fig. 1͑d͒ u͑q͒ typically montonically decreases 4kF͑␻͒ϳ ͉ ͑ ͉͒2͉␻ ␻ ͉4Kc−1 ͑ ͒ SQ ͚ u 4kF ± I , 29 with increasing ͉q͉. Thus, the smaller the momenta, the larger ± ␹ ͑ ␻͒ the contribution from 0 q, . ␻ which exhibits frequency dependence centered around ± I. It turns out ͑somewhat reminiscently of the Coulomb drag Ϸ Collecting the q 0,2kF and 4kF contributions we have, case͒ that the qϷ0 piece does not contribute to the dephas-

035336-6 NONEQUILIBRIUM CHARGE NOISE AND DEPHASING… PHYSICAL REVIEW B 75, 035336 ͑2007͒

spin-incoherent Luttinger liquid with finite current I. Both ͑␻͒ ␶ the noise SQ and the dephasing time ␸ are sensitive to the density correlations in the strongly interacting 1D system, and these correlations qualitatively change when the temperature T becomes of order the magnetic exchange energy J. In the noise, this leads to a loss in the frequency ␻Ϸ␻ ϵ␲ response near I /2 I/e, corresponding to a loss of the 2kF component of the density correlations. In the dephasing rate, a non-monotonic dependence on temperature should be observed with a transition around TϳJ, again due to the FIG. 3. Schematic of the temperature dependence of the dephas- loss of the 2kF component of the density correlations. ing time of a qubit as described in the text. The geometry of the So far, there are experimental indications pointing to the proposed experiment is shown in Fig. 1. The temperature depen- realization of the SILL in quantum wires with low electron ␶ ͑ ͒ dence of the dephasing time ␸ is given by Eq. 35 , and an ap- density,41 but these data are only preliminary and restricted ͑ ͒ proximate form by Eq. 36 . The constant c in the exponential is to relatively few particle numbers. At present extremely high ͑ ͒ ␻ O 1 and 0 is a high frequency cutoff in the charge sector. The quality quantum wires with low electron density appear to be figure assumes 1/4ϽK Ͻ1 and K =1. c s a very promising candidate for observing spin-incoherent effects and there are now many falsifiable theoretical predic- ing due to the same phase space restrictions imposed by the tions to be put to the test. We hope those detailed here will linear dispersion that kill this contribution to the drag be- soon be tested experimentally. tween quantum wires.23 This is easily seen from Eq. ͑12͒ ␹qϷ0͑ ͒ϵ which shows that 0 q,−vdq 0 since vc vd ensuring the delta function is zero for all finite q. This leaves the two ACKNOWLEDGMENTS ␶ dominant contributions to ␸ coming from the 2kF and 4kF We thank L. Balents for enlightening discussions. This pieces work was supported by NSF Grant numbers PHY99-07949, 1 1 1 DMR04-57440, DMR03-34499, and the Packard Founda- Ϸ + . ͑36͒ tion. G.A.F. was also supported by the Lee A. DuBridge ␶ ␶2kF ␶4kF ␸ ␸ ␸ Foundation. We gratefully acknowledge the hospitality of the We obtain the temperature dependence of each of these Aspen Center for Physics where part of this work was carried pieces from Eqs. ͑14͒ and ͑27͒ together with Eq. ͑35͒.We out. ␻ now assume small currents such that kT I. We find 2 APPENDIX: DERIVATION OF DEPHASING RATE FOR A 1 J1 ϳ͉ ͑ ͉͒2␳2ͩ ͪ Ks+Kc−1 −cT/J ͑ ͒ QUANTUM DOT u 2kF T e 37 ␶2kF 0 ␻2 2 ␸ m 0a A problem completely analogous to the dephasing of a and qubit as discussed in Sec. IV is that of dephasing of a quan- 42,43 1 tum dot. Following Levinson we outline here how one ϳ͉ ͑ ͉͒2␳2 4Kc−1 ͑ ͒ u 4kF T . 38 arrives in that case at the expressions for the dephasing rate ␶4kF 0 ␸ used in Sec. IV. We assume the Hamiltonian for the problem

The temperature dependence of ␶␸ is shown schematically is given by H=Helec+HQD+Hint where Helec is given by Eqs. Ͻ Ͻ ͑1͒ and ͑2͒, in Fig. 3 for 1/4 Kc 1. In this regime, the dephasing time may exhibit non-monotonic behavior with temperature in a H = ⑀ c†c ͑A1͒ way somewhat reminiscent of the Coulomb drag resistance QD 0 between two parallel quantum wires.23 For TՇJ, the dephas- and ing time may actually increase as the temperature increases. e2 This is counter to the naive expectation that heating up the † ͩ ͪ͵ ͑ ͒␦␳͑ ͒ ͑ ͒ Hint = c c dxu x x , A2 system ͑bath͒ is likely to decrease the coherence time be- 2C cause presumably the fluctuations are increasing. In fact, the where as before e is the charge of the electron and C is the fluctuations the qubit experiences decrease as T→J from capacitance of the quantum dot. From Eq. ͑A2͒ it is clear that below because the 2k fluctuations that dominate the noise F the effect of the interaction of the electrons in the wire with ͓ ͉ ͑ ͉͒2͑ J1 ͒2 Ͼ͉ ͑ ͉͒2͔ assuming u 2kF 2 2 u 4kF in this temperature ⑀ m␻0a the quantum dot is to shift the level 0 of the dot by տ e2 range are washed out for T J leaving only the weaker 4kF ͐ ͑ ͒␦␳͑ ͒ ␦␳͑ ͒ 2C dxu x x . Indeed, if x depends on time, then the contributions. An observation of this behavior is strong evi- quantum dot level will “jiggle” and this will lead to dephas- dence for the spin-incoherent Luttinger liquid. ing, or decoherence.44,45 Below we define and calculate the ␶ V. CONCLUSIONS dephasing rate ␸ as a function of temperature. It is conceptually convenient to discuss the decoherence We have studied the nonequilibrium noise spectrum and in terms of density matricies. We will assume the quantum 46 ͉␺͘ ␣ ͉ ͘ the dephasing rate in a small device in close proximity to a dot has been initially prepared in a state = 0 0

035336-7 GREGORY A. FIETE AND MARKUS KINDERMANN PHYSICAL REVIEW B 75, 035336 ͑2007͒

␣ ͉ ͘ ͉ ͘ϵ † ͉ ͘ ͑ ͒ + 1 1 with 1 c 0 . This leads to the density matrix We thus approximate the time evolution of W t ͑ ͒ ͑ ͒ Ϸei Helec+HQD tWe−i Helec+HQD t =eiHelectWe−iHelect, effectively ␳ ͉␺͗͘␺͉ ͑ ͒ ˆ = . A3 keeping the lowest order corrections in W. In a cumulant ͑ ͒ Coherence in a quantum system is manifest in nonzero off expansion of Eq. A5 the leading contribution is ⑀ ⌽͑ ͒ diagonal elements of the density matrix. At time t=0 we ͗c͑t͒͘Ϸ͗c͑0͒͘e−i 0te− t , ͑A6͒ ͗ ͑ ͒͘ ␣ ␣* have c 0 = 1 0 0, so the initial state is coherent. We ⌽͑ ͒ 1 ͐ ͐ ͑ ͒ ͑ ͒ ͗ ͑ ͒ ͑ ͒͘ wish to compute the decay of the off-diagonal density matrix with t = 2 dt1 dt2K t1 −t2 and K t1 −t2 = W t1 W t2 , elements ͑from density fluctuations in the 1D system͒ given where we have used the fact that the correlator is a function by ͗c͑t͒͘, where c͑t͒=eiHtce−iHt. By taking the time deriva- only of the time difference. By making the change of vari- d ͑ ͒ ⌽͑ ͒ ͑ ͒ ͑⑀ ͑ ͒͒ ͑ ͒ ables y1 =t1 −t2 and y2 = t1 +t2 /2, we can express t as tive of this equation one finds dtc t =−i 0 +W t c t , where e2 ͑ ͒ iHt −iHt ϵ ͐ ͑ ͒␦␳͑ ͒ t W t =e We and W 2C dxu x x . This can be inte- 1 ⌽͑ ͒ ͵ ͑ ͒ ͑ ͒ grated to yield t = t dy1K y1 . A7 2 −t t −i⑀ t −i͵ dtЈW͑tЈ͒ The function K͑t͒ has some characteristic cutoff time ␶ such ͑ ͒ ͑ ͒ 0 ͑ ͒ c c t = c 0 e Tte . A4 0 ͑ ͒Ϸ Ͼ␶ that K t 0 when t c. Thus, for long times the integral Therefore, we obtain the exact expression43 above can be extended to infinity defining a dephasing time ␶␸, t −i⑀ t −t/␶ −i⑀ t −i͵ dtЈW͑tЈ͒ ͗c͑t͒͘Ϸ͗c͑0͒͘e 0 e ␸ ͑A8͒ ͗ ͑ ͒͘ 0 ͗ ͑ ͒ ͘ ͑ ͒ c t = e c 0 Tte . A5 0 with We now assume that the interaction u͑x͒ between the quan- ϱ 1 1 tum dot and the 1D system is sufficiently weak that there are = ͵ dtK͑t͒. ͑A9͒ ␶ negligible back-action effects of the dot on the 1D system. ␸ 2 −ϱ

1 C. Beenakker and C. Schönenberger, Phys. Today 56,37͑2003͒. 20 G. A. Fiete and L. Balents, Phys. Rev. Lett. 93, 226401 ͑2004͒. 2 Y. M. Blanter and M. Büttiker, Phys. Rep. 336,1͑2000͒. 21 G. A. Fiete, J. Qian, Y. Tserkovnyak, and B. I. Halperin, Phys. 3 L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne, Phys. Rev. Rev. B 72, 045315 ͑2005͒. Lett. 79, 2526 ͑1997͒. 22 G. A. Fiete, K. Le Hur, and L. Balents, Phys. Rev. B 72, 125416 4 R. de Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bu- ͑2005͒. nin, and D. Mahalu, Nature ͑London͒ 389, 162 ͑1997͒. 23 G. A. Fiete, K. Le Hur, and L. Balents, Phys. Rev. B 73, 165104 5 M. Reznikov, R. de Picciotto, T. G. Grifﬁths, M. Heiblum, and V. ͑2006͒. Umansky, Nature ͑London͒ 399, 238 ͑1999͒. 24 M. Kindermann and P. W. Brouwer, Phys. Rev. B 74, 125309 6 R. Hanbury-Brown and R. Q. Twiss, Nature ͑London͒ 178, 1447 ͑2006͒. ͑1956͒. 25 M. Kindermann, P. W. Brouwer, and A. J. Millis, Phys. Rev. Lett. 7 M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. 97, 036809 ͑2006͒. Holland, and C. Schönenberger, Science 284, 296 ͑1999͒. 26 P. Kakashvili and H. Johannesson, cond-mat/0605062 ͑unpub- 8 W. D. Oliver, J. Kim, R. C. Liu, and Y. Yamamoto, Science 284, lished͒. 299 ͑1999͒. 27 G. A. Fiete, Phys. Rev. Lett. 97, 256403 ͑2006͒. 9 B. Trauzettel, I. Saﬁ, F. Dolcini, and H. Grabert, Phys. Rev. Lett. 28 K. A. Matveev, A. Furusaki, and L. I. Glazman, cond-mat/065308 92, 226405 ͑2004͒. ͑unpublished͒. 10 A. V. Lebedev, A. Crépieux, and T. Martin, Phys. Rev. B 71, 29 V. V. Cheianov, H. Smith, and M. B. Zvonarev, Phys. Rev. A 71, 075416 ͑2005͒. 033610 ͑2005͒. 11 M. Sindel, W. Hofstetter, J. vonDelft, and M. Kindermann, Phys. 30 J. Voit, Rep. Prog. Phys. 58, 977 ͑1995͒. Rev. Lett. 94, 196602 ͑2005͒. 31 T. Giamarchi, Quantum Physics in One Dimension ͑Clarendon 12 D. V. Averin and Y. V. Nazarov, Phys. Rev. B 49, 2951 ͑1994͒. Press, Oxford, 2004͒. 13 M. H. Pedersen, S. A. van Langen, and M. Büttiker, Phys. Rev. B 32 F. Haldane, J. Phys. C 14, 2585 ͑1981͒. 57, 1838 ͑1998͒. 33 The dephag of the excitations within the Luttinger liquid itself has 14 K. A. Matveev, Phys. Rev. Lett. 92, 106801 ͑2004͒. been discused in I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, 15 K. A. Matveev, Phys. Rev. B 70, 245319 ͑2004͒. Phys. Rev. Lett. 95, 046404 ͑2005͒;K.LeHur,ibid. 95, 16 G. A. Fiete, cond-mat/0611597 ͑unpublished͒. 076801 ͑2005͒. The case of excitations in a spin-incoherent Lut- 17 M. M. Fogler, Phys. Rev. B 71, 161304 ͑2005͒. tinger liquid is discussed by K. Le Hur in Phys. Rev. B 74, 18 V. V. Cheianov and M. B. Zvonarev, Phys. Rev. Lett. 92, 176401 165104 ͑2006͒. ͑2004͒. 34 Throughout this paper we neglect any back action of the gate or 19 V. V. Cheianov and M. B. Zvonarev, J. Phys. A 37, 2261 ͑2004͒. dot on the quantum wire. In principle, the gate and dot can

035336-8 NONEQUILIBRIUM CHARGE NOISE AND DEPHASING… PHYSICAL REVIEW B 75, 035336 ͑2007͒

provide a local potential disturbance in the wire that can pin the 39 M. Büttiker and A. M. Martin, Phys. Rev. B 61, 2737 ͑2000͒. Wigner crystal at sufﬁciently small source-drain voltages. How- 40 S. Pilgram and M. Büttiker, Phys. Rev. Lett. 89, 200401 ͑2002͒. ever, estimating the energy scale for this is not straightforward at 41 H. Steinberg, O. M. Auslaender, A. Yacoby, J. Qian, G. A. Fiete, it relies on a renormalization group analysis as discussed in Y. Tserkovnyak, B. I. Halperin, K. W. Baldwin, L. N. Pfeiffer, Ref. 31. and K. W. West, Phys. Rev. B 73, 113307 ͑2006͒. 35 M. Pustilnik, M. Khodas, A. Kamenev, and L. I. Glazman, Phys. 42 Y. Levinson, Phys. Rev. B 61, 4748 ͑2000͒. Rev. Lett. 96, 196405 ͑2006͒. 43 Y. Levinson, Europhys. Lett. 39, 299 ͑1997͒. 36 M. Pustilnik, E. G. Mishchenko, L. I. Glazman, and A. V. An- 44 G. A. Fiete and E. J. Heller, Phys. Rev. A 68, 022112 ͑2003͒. dreev, Phys. Rev. Lett. 91, 126805 ͑2003͒. 45 A. Stern, Y. Aharonov, and Y. Imry, Phys. Rev. A 41, 3436 37 M. Pustilnik, Phys. Rev. Lett. 97, 036404 ͑2006͒. ͑1990͒. 38 S. Sachdev, Quantum Phase Transitions ͑Cambridge University 46 D. Sprinzak, E. Buks, M. Heiblum, and H. Shtrikman, Phys. Rev. Press, Cambridge, 1999͒. Lett. 84, 5820 ͑2000͒.

035336-9