Electronic and photonic counting statistics as probes of non-equilibrium quantum dynamics

Abstract

We analyse the full counting statistics (FCS) of photons flowing in and out of a microwave cavity coupled to a voltage-biased Josephson junction. Tunnelling of Cooper pairs generates a coherent flow of photons into the cavity whilst at the same time photons can also leak out incoherently. We use a very general unitary transformation method to demonstrate that there is a simple connection between the FCS of the charges and the photons in the long time limit, revealing that all the cumulants of the coherent and the incoherent processes match in that limit. We also explore some of the interesting features in the counting statistics of the charges and photons which arise from the strongly nonlinear dynamics of the system. These include very narrow distributions associated with the emergence of coherent transport and regimes where counting of either an odd or an even number of photons leaving the cavity can result in strongly non-classical cat states within the cavity. where the renormalized Josephson energy is defined as = -D E E e J J 2 0 2 D  E E z J z e 4 J J B 1 2 0 1 0 2 0 2[ ( ) ] with z 1 ;1.841), though it rises abruptly at the bifurcation, before slowly dropping away again for even larger E J . In terms of the Figure 2. (a) Distributions for coherent photon transfer into the cavity, P N t ,( ), and incoherent emission from the cavity, P(N, t), at different times. (b) Behaviour of P(N, t) below and above the bifurcation at = E E J J B together with the corresponding Poissonian distributions (dashed); γt=80 (except for = E E 1.6 J J Bwhere γt=60 for clarity). (c) Large deviation functions (for γt=80); the Poissonian case is shown as a dashed line. Δ 0 =0.5 throughout. 7 If instead we use just the part of ρ(0) which is diagonal (in the number state basis) in equation (15) to define our moment generating function, then the counting distribution we obtain remains positive. This is because doing so amounts to defining a generating function for a counting distribution corresponding to two-point projective measurements of the photon number (see section II of [40]).

Thermodynamic uncertainty relations for coherently driven open quantum systems

“…) to obtain ∂ t ρt = L[s]ρ t ≡ −i [ω a † a + Ω (a + a † ), ρt ] + γ(n + 1) (e s aρ t a † − {a † a, ρt }/2)+ γn (e −s a † ρt a − {aa † , ρt }/2). (B.1)As shown in Ref [68],. the Liouvillian L[s] is similar to the transformed LiouvillianL [s]ρ t ≡ −iω [a † a, ρt ] − iΩ (e −s/2 a + e s/2 a † )ρ + iΩ ρ(e s/2 a + e −s/2 a † ) + γ(n + 1) (aρ t a † − {a † a, ρt }/2) + γn (a † ρt a − {aa † , ρt }/2), (B.2)where the counting field now appears in the driving terms.…”

Nonequilibrium phase transition in a single-electron micromaser

“…Micromasers allow for detailed investigations of lightmatter interactions in quantum cavity electrodynamics [1][2][3]. A micromaser consists of a high-quality photon cavity, which is driven by excited two-level systems, such as a beam of atoms [4,5], a superconducting singleelectron transistor [6][7][8][9], or a double quantum dot [3,[10][11][12][13][14]. Several experiments [15][16][17][18][19][20][21][22][23][24][25][26] with micromasers and their optical counterparts, microlasers, have revealed a variety of intriguing quantum phenomena such as quantum collapse and revival [16], the generation of nonclassical atom statistics [20,22,25], and non-destructive single-photon detection [19].…”

Josephson photonics with simultaneous resonances

“…Furthermore, the energy transferred by tunneling Cooper pairs into microwave modes can be tracked by monitoring either the resulting dc current or the microwaves leaking out of the circuit 1 . Recent experimental, [1][2][3][4][5][6] and theoretical work [7][8][9][10][11][12][13][14][15][16][17][18] has explored a wide range of ways in which JJ-cavity systems can be used to generate non-classical microwave states.…”