Theory of interaction-dependent instability in quantum detection by means of a Luttinger liquid tunnel junction: A rigorous theorem

Abstract

The low-temperature regime of charge-qubit decoherence due to its Coulomb interaction with electrons tunneling through Luttinger liquid quantum-point contact (QPC) is investigated. The study is focused on quantum detector properties of Luttinger liquid QPC. Earlier results on related problems were approximate, up to the second order in small electrostatic coupling between chargequbit and QPC. However, here it is shown that in low-(and zero-)temperature limit the respective perturbative decoherence-and acquisition of information timescales both tend to diverge, thus, shadowing a true picture of low-temperature quantum detection for such quantum systems. Here it is shown, that one can successfully circumvent these difficulties in order to restore complete and exact picture of low-temperature decoherence and quantum detection for charge-qubit being measured by arbitrary Luttinger liquid QPC. To do this, here I prove two general mathematical statements (S-theorem and S-lemma) about exact re-exponentiation of Keldysh-contour ordered T-exponent for arbitrary Luttinger liquid tunnel Hamiltonian. The resulting exact formulas are believed to be important in a wide range of those Luttinger liquid problems, where real-time quantum field dynamic is crucial. As the result, decoherence-and acquisition of information time-scales as well as QPC quantum detector efficiency rate are calculated exactly and are shown to have a dramatic dependence on repulsive interaction between electrons in 1D leads of QPC. In particular, it is found that at temperatures close to zero there exists a certain well-defined threshold value g ≈ gcr(T ) of Luttinger liquid correlation parameter g (0 < g ≤ 1) which serves as a sharp boundary between region of good (or even perfect) quantum detection at g < gcr and the region of quantum detection breakdown for g > gcr. Moreover, discovered abrupt decrease of QPC quantum detector efficiency Q with the increase of g in the close vicinity of value gcr represents a fingerprint of interactiondependent instability of all the quantum detection procedure for any Luttinger liquid QPC quantum detector at definite low enough temperatures Tcr(g). The reasons behind these effects are discussed. Also, it is shown that such the low-temperature detection instability effect is able to explain a large unclear mismatch between expected and observed decoherence timescales in two recent experiments ( J.Gorman, D.G.Hasko, D.A.Williams, Phys.Rev.Lett., 95, 090502, (2005) and K.D.Petersson, J.R.Petta, H.Lu, A.C.Gossard, Phys.Rev.Lett., 105, 246804 (2010) ) on charge-qubit quantum dynamics.PACS numbers:

Tunneling maps of interacting electrons in real time: anomalous tunneling in quantum point-contacts beyond the steady state regime

“…The problem of the effective quantum detection of qubit quantum states lies in the core of a contemporary progress in quantum information science and quantum computers design [1][2][3][4][5][6]. Especially, this concerns chargequbits constructed as spatial electron states being delocalized within two tunnel-connected metallic droplets which form a quantum double-dot in the regime of Coulomb blockade [5,6,[11][12][13]. In the latter case of our interest a biased quantum-point contact (QPC) plays the role of quantum detector [6,11] of electron states on the double dot if one couples the former capacitively to a given charge qubit [12,13].…”

“…Especially, this concerns chargequbits constructed as spatial electron states being delocalized within two tunnel-connected metallic droplets which form a quantum double-dot in the regime of Coulomb blockade [5,6,[11][12][13]. In the latter case of our interest a biased quantum-point contact (QPC) plays the role of quantum detector [6,11] of electron states on the double dot if one couples the former capacitively to a given charge qubit [12,13]. In this case electron states on the double-dot are easy to prepare and to manipulate by external gates [1][2][3][4] though the problem of decoherence due to back-action from quantum detector remains challenging in such type of charge-qubit design [11][12][13].…”

“…Here the best situation for qubit states preparation and detection can be realized when one neglects the internal quantum dynamics of charge qubit due to very slow Rabi oscillations of the electron quantum state on the double dot with non-zero inter-dot tunneling in comparison with the characteristic time scales of qubit decoherence and acquisition of information by the QPC quantum detector [6,11]. However, even in the experimental situations being that ideal one faces several fundamental problems which can considerably smear or even misrepresent the information about the qubit decoherence timescales and, hence, about the true qubit quantum states being detected by means of given QPC [11][12][13]. In particular, one factor here is a well-known Anderson orthogonality catastrophe due to Coulomb electrostatic interaction between the electron delocalized on the double dot and the electrons tunneling through the QPC quantum detector [6,11].…”

“…However, even in the experimental situations being that ideal one faces several fundamental problems which can considerably smear or even misrepresent the information about the qubit decoherence timescales and, hence, about the true qubit quantum states being detected by means of given QPC [11][12][13]. In particular, one factor here is a well-known Anderson orthogonality catastrophe due to Coulomb electrostatic interaction between the electron delocalized on the double dot and the electrons tunneling through the QPC quantum detector [6,11]. This effect leads to a fast decoherence of any initially prepared charge-qubit state superposed from its states |1 and |2 on the dot 1 and on dot 2, correspondingly, as compared to the characteristic timescales of information acquisition by the statistics of tunneling events for given QPC [6,11].…”

“…In particular, one factor here is a well-known Anderson orthogonality catastrophe due to Coulomb electrostatic interaction between the electron delocalized on the double dot and the electrons tunneling through the QPC quantum detector [6,11]. This effect leads to a fast decoherence of any initially prepared charge-qubit state superposed from its states |1 and |2 on the dot 1 and on dot 2, correspondingly, as compared to the characteristic timescales of information acquisition by the statistics of tunneling events for given QPC [6,11]. Another basic effect for one-dimensional QPCs is Coulomb repulsion between the electrons in the one-dimensional geometry of quantum wires [7][8][9], which leads to a strong suppression of tunneling density of states at the edges of Luttinger liquid QPC electrodes in the close vicinity of the Fermi energy [9,10].…”