F Montel

Cendrine Faivre-Moskalenko

Martin Castelnovo
email: martin.castelnovo@ens-lyon.fr

Dynamical DNA accessibility induced by chromatin remodeling and protein binding

come

Dynamical DNA accessibility induced by chromatin remodeling and protein binding

Introduction -Nucleosomes represent the first degree of organization of DNA within eukaryotic chromatin, in which the nucleic acid is wrapped roughly two times around a proteic core made of histones [1]. The wrapping geometry of the nucleosome constitutes on the one hand a physical barrier to DNA translation and replication, and on the other hand a way of regulating gene expression. One of the regulation strategy observed within cells is the use of remodeling factors [2]. These protein complexes are indeed able to move nucleosomes along DNA templates upon hydrolyzing ATP, and they are therefore able to generate transient windows of DNA accessibility for protein binding (transcription factors or activators for example). While chromatin remodelers have been identified almost ten years ago, their mechanisms of action are still subject of intense researches [3][4][START_REF] Mobius | Proc. Nat. Acad. Sci. USA[END_REF]. In particular, results about their collective effects in the modulation of nucleosome density at large scale are more scarce [START_REF] Parmar | [END_REF].

In this work, we show how the cooperative action of chromatin remodeling and protein binding may influence nucleosome density and DNA accessibility at large scale by mimicking the dynamics induced by remodeling factors from the swi2/snf2 family (for example RSC remodeling factor). Starting from well-positioned nucleosomal arrays, as the one observed in vivo after DNA replication, we use coarse-grained numerical simulations in order to quantify DNA accessibility induced by the presence of remodeling factors and protein binding, the number of nucleosomes being constant. The main input in the simulation is the typical mobilization rule for individual nucleosome. These rules are based on recent Atomic Force Microscopy observations on small oligo-nucleosomal templates (di-and tri-nucleosomes) remodeled by [7]. In this work, it has been observed that an individual remodeling event induces a displacement of nucleosome from its initial position towards either another nucleosome or the end of DNA template without any observable intermediates. Using this simple rule as the basic ingredient of the simulations, it is possible to investigate the structuration of nucleosomal density in the presence of remodeling factors and protein binding, and consequently the modulation of DNA accessibility. Our simulations identify the following features: (i) the remodeling by complexes of swi2/snf2 family of nucleosomal array induces the coexistence of high and low nucleosomal density regions; (ii) the lifetime of DNA accessibility is strongly inhomogeneous along DNA template, although no sequence effect is explicitly considered ; (iii) the simultaneous presence of remodelers and protein binding allows to modulate spatially the DNA accessibility. ordered with a linker length L k , therebye imposing a fixed nucleosome density. The total length of DNA is therefore L tot = N L c + (N + 1)L k , where L c is the DNA length covered by a single nucleosome. For the sake of simplicity, we choose the canonical value L c = 147bp.

Simulation rules -We consider an oligomeric array composed of N nucleosomes initially

Choosing linker lengths between 20bp and 50bp, the nucleosome occupancy is modulated between 75% and 88% [8]. This choice of fixed density allows to focus on events occuring between DNA replication phases, unlike Parmar et al. who investigated the evolution of overall nucleosome density upon nucleosome deposition starting from naked DNA [START_REF] Parmar | [END_REF]. The general algorithm used in the simulation is the following: at each simulation step k nucleosomes (typically k = N ) are chosen randomly and a local motion rule (to be described below) is applied successively on these k nucleosomes. Only moves avoiding sterical clash are effectively performed. This elementary step is then iterated. Elementary runs are then reproduced in order to reach statistical significance of the results. This versatile design of the simulation allows addressing the comparison of different local motion rules. The results presented in this work were obtained by two different elementary motion rules: (i) RSC -like mobilization (swi2/snf2 family), for which the nucleosome is moved processively with equal probability to its right or to its left until it reaches either another nucleosome or the end of DNA template, and (ii)thermal -like mobilization, for which the nucleosome travels a fixed distance to its left or to its right with equal probability. For this last mobilization scenario, the system is then equivalent to a Tonk's gas. In our study, we chose a thermal jump size of 10bp, which is consistent with rotational positioning of nucleosomes [9]. The elementary simulation step during which k motion of nucleosomes are tried is considered as our time unit. Therefore our results will be valid for time larger than this elementary time scale.

Following Narlikar et al. [10], we consider that the rate limiting step associated to nucleosome remodeling is the sliding of the nucleosome. Note that the quantitative comparison of time scales for both mobilization scenarios is not addressed within the present work since it is strongly correlated with the rate of individual mobilization events. This question goes beyond the scope of our investigations. For the sake of simplicity, we chose the same rate for RSC and thermal mobilization since it allows to highlight differences in nucleosomal patterns. Additionally, within our simulation design, detailed balance for the different rates is expected to breakdown, because each individual motion of RSC-mobilized nucleosomes is performed thanks to the consumption of external energy (ATP). This external forcing of nucleosome motion breaks the detailed balance, as it has been already suggested and discussed in the literature in the context of molecular motors or self-assembly modeling [11][12][13].

In order to analyze the evolution of DNA accessibility during the simulation, we used probes of variable size (typically 10bp) located at the dyad of every nucleosomes in the starting configuration. Each probe will produce a unit value whenever it is not covered by a nucleosome, and the probe status (free or occupied) is updated at each simulation step. This permits to monitor precisely the evolution of accessibility. For the investigation of protein binding on oligo-chromatin template, we chose the simplest rule for binding in order to analyze the asymptotic stationary state of the simulations: after an initial period of RSC-mobilization in order to reach stationary state before binding, the protein will bind at once and irreversibly at the first simulation step when the site is free of nucleosome. The simulation is then continued for sufficient time in order to reach a new stationary state in the presence of the protein. implies that the bulk average nucleosome occupancy is always larger in thermal mobilization as compared to RSC mobilization. The two previous estimations are in good agreement with the simulation results. Note that in the case of RSC mobilization, the expression for ρ

Results

(∞) nuc,RSC
is valid whenever L k < L c , which implies the presence of a finite size plateau. The two previous estimations for the bulk average nucleosome occupancy are valid up to 1/N order.

The range of nucleosome accumulation for RSC mobilization at the boundaries is estimated by the size of the nucleosome depleted region. In the rescaled units of figure 2, this range is therefore L border /L tot = L k /(L c + L k ). This estimation is again in good agreement with the simulations. The range of boundaries effect are opposite for thermal mobilization : it decreases with the linker length. This effect has already been discussed in the literature [START_REF] Mobius | Proc. Nat. Acad. Sci. USA[END_REF].

The lifetime of DNA accessibility as measured by probes of given sizes along the nucleosomal array provides informations about the dynamics of remodeling in the stationary state. the introduction section of this work, the quantitative comparison of accessibility time scales is not possible within this work since it depends on the particular choice of elementary mobilization rates. Interestingly, RSC mobilization shows another striking feature, namely the relative inhomogeneity of lifetime distributions close to the boundaries of DNA template:

indeed, the border regions identified in the average nucleosome occupancy profiles shows large variations of lifetime distribution. In particular, the average lifetime has a low value close to DNA boundary, reflecting the accumulation of nucleosomes at the boundaries, and it increases rapidly towards the bulk of nucleosomal array. Remarkably, it reaches a maximum at roughly (N + 1)L k then it decreases sharply towards a constant value for the bulk sites. In other words, there is a highly localized region of DNA accessibility enhancement, in which the average lifetime is roughly doubled as compared to bulk accessibility. This trend is also observed for the distribution of lifetime accessibility, as it is evidenced in figure 3b.

This result is understood by analyzing the motion of the depleted region with respect to the considered site. When a given site starts to be accessible, the depleted region overlap this site over the length a of the probe. The accessibility event stops whenever this overlapping vanishes by diffusion of the depleted region. For bulk sites, accessibility events can be nucleated and finished at similar or opposite location, as it is illustrated in figure 3c. On the contrary, for sites close to the DNA boundaries, accessibility events are necessarily nucleated and finished at the same locations. As a consequence, accessibility events might last longer for these border sites than for bulk sites. The boundary between these two regions is extremely sharp, as it is reflected by the variation of average lifetime accessibility.

It is possible to estimate the shape of the average accessibility lifetime profile by considering the mean first passage time of random walker on a line with two adsorbing boundary conditions. It is a standard result that the mean first passage time associated with a 1D random walker of diffusion coefficient D , starting at position x 0 and confined between two adsorbing walls located at x = 0 and x = L is τ M F P T = x 0 (L-x 0 ) 2D [START_REF] Gardiner | Handbook of stochastic methods: for physics, chemistry and natural sciences[END_REF][START_REF] Gitterman | [END_REF]. For the bulk sites, the average accessibility lifetime scales therefore as t bulk ∼ L c ((N + 1)L k -L c )/2D RSC , with

D RSC = L 2
c /τ RSC with τ RSC typical mobilization time by RSC. This time is independent on the position in the template. In the region close the border of the template at position X, the average accessibility lifetime is the mean first passage time of random walk between one adsorbing and one reflecting wall roughly separated by X, and starting at a distance L c from the adsorbing wall. Using mirror symmetry in order to solve this problem, this time is equivalent to the mean first passage time of random walker between two adsorbing walls separated by 2X. Therefore it is then given by t border ∼ L c (2X -L c )/2D RSC . These formulas explain quantitatively the emergence of a localized region with greater average accessibility than the bulk for (N + 1)L k /2 < X < (N + 1)L k . In particular the maximal average accessibility lifetime is predicted to be roughly twice the lifetime for bulk sites, in agreements with the simulation results.

Finally, in order to illustrate the effect of protein binding on DNA accessibility, we allowed some sites to be bound by proteins of fixed size. This binding has two consequences on DNA demonstrate that the induction of swi/snf remodeler induces mean nucleosome occupancy profiles very similar to the one produced by our simulation [START_REF] Tolstorukov | Proc. Nat. Acad. Sci. USA[END_REF]: the nucleosome occupancy increases as the so-called transcription start site (TSS) and its particular nucleosome architecture (NFR) is approached for wild-type conditions, while this increase is strongly reduced if remodelers are inhibited. In addition, we predict within our simulation a localized region for which the accessibility lifetime is roughly doubled as compared to the other regions.

Interestingly, this last feature is quite a general result from 1D-diffusion of an accessibility windows confined between reflecting boundaries. The only requirement in order to observe a region of accessibility enhancement close to reflecting boundaries is that the size of the accessibility windows W is larger than the size of the probe a. As a consequence, the accessibility lifetime of sites located in the range [W/2; W ] are larger than bulk sites in the range

[W + a; L -W -a].
In order to progress further towards the global understanding of DNA accessibility within chromatin, both sequence effects and protein binding kinetics are to be taken into account in future works. The former effect might essentially influence the diffusion of the nucleosome depleted region on a bumpy energetic landscape, while the latter might influence the lifetime of accessibility patterns. The present results strongly suggest that simulations of the dynamics are efficient tools in order to investigate such complex molecular scenarios.

FIG. 1 :

 1 FIG. 1: Comparison of nucleosomal profiles after thermal and RSC mobilization. (a) Average nucleosomal DNA, in the common starting configuration (left), after 2000 steps of thermal (center) and RSC (right) mobilization. Snapshots of nucleosomal DNA at different times for thermal (b) and for RSC (c) mobilization. Parameters for this figure are N = 20, L k = 50bp, 2000 runs. Cartoons have illustrative purposes for typical stationary configurations for thermal and RSC mobilization, but not at proper scale.

FIG. 2 :

 2 FIG. 2: (Color online) Dependence of average nucleosomal profile in N and L k for thermal (left panel) and RSC (right panel) mobilization. The four curves on each panel are associated with parameters: N = 20 and L k = 50bp (dark thick line), N = 20 and L k = 20bp (light thick line), N = 40 and L k = 50bp (bottom thin line), N = 50 and L k = 20bp (upper thin line) .Other common parameters: 2000 steps, 2000 runs.

The accessibility lifetimes is shown for RSC mobilization in figure 3 .FIG. 3 :

 33 FIG. 3: DNA accessibility lifetime upon RSC mobilization at discrete sites hidden by nucleosomes in the starting configuration. (a) Average accessibility lifetime for N = 6 (lower blue line with circles), N = 20 (middle green line with circles), N = 40 (upper red line with circles). The dotted lines correspond to the average accessibility lifetime for thermal mobilization for N = 20 (light cyan) and N = 40 (dark purple). Other common parameters: L k = 50bp, 2000 steps, 2000 runs. (b) Accessibility lifetime distribution in logarithmic color scale for N = 20. The green curve correspond to the average accessibility lifetime. (c) Cartoons for N = 6 showing only one entry/exit (yellow star) for accessibility event (top, red light stick), and two entries/exits (down, dark blue stick). The configurations allowing accessibility of sites are highlighted by the change of color of the sticks (light yellow sticks).

FIG. 4 :

 4 FIG.4: Comparison of nucleosomal DNA and average accessibility lifetime after protein bindingat different locations L prot . (a) L prot = L tot /2, (b) L prot = L tot /3, (c) L prot = L tot /4, (d) L prot =L tot /5. For each panel, the top (blue) curve is the nucleosomal DNA, while the lower (green) curve is the average accessibility lifetime. Thin (red) dotted lines represent the average accessibility lifetime before binding. Other common parameters: N = 40, L k = 50bp, 4000 steps, 500 runs.


Gilles Saint

Gilles Saint-Paul

Can active labor market policy be counter-productive?

Keywords:

We study active labor market policies (ALMP) in a matching model. ALMPs are modelled as a subsidy to job search. Workers di¤er in their productivity, and search takes place along an extensive margin. An additional job seeker a¤ects the quality of unemployed workers. As a result, the Hosios conditions are no longer valid. To replicate the optimum the worker share in bargaining must exceed the Hosios level, and one must impose a tax on job search activity. The coalition in favor of ALMP is also studied.

Introduction

This paper studies the e¤ect of active labor market policies (ALMP) in a Mortensen-Pissarides style matching model. ALMPs are modelled as a subsidy to job search, and it is assumed that search activity is observed. A key feature of the model is that workers di¤er in their productivity, and that search takes place along an extensive margin. The model is used to study the e¤ect of ALMP on the equilibrium, on aggregate welfare, and, equally importantly, on the distribution of welfare across worker types (productivity levels) and current labor market status (employed vs. unemployed).

It is shown that in addition to the usual job search externality, there is a "quality" externality. As search is not directed, an additional job seeker a¤ects the average quality of the pool of unemployed, in addition to the job …nding rate. As a result, the usual "Hosios" conditions for an e¢ cient outcome -that the bargaining share of workers match their elasticity in the matching function -are no longer valid 1 . For an e¢ cient outcome, the decentralized equilbrium conditions must match the optimal ones for both the job creation margin of …rms and the job search decision of workers, and these two conditions cannot be matched with a single instrument. It is shown, paradoxically, that to replicate the optimum one must select a worker share in bargaining which is larger than their elasticity in the matching function, and at the same time one must impose a tax on job search activity.

Clearly, this prediction does not validate the view that ALMPs are a desirable policy tool. The reason is that they raise workers'outside option in bargaining, thus contributing to wage pressure, while at the same time reducing the average quality of job seekers. The optimal policy outlined above delivers an improved quality of job seeker, due to the search tax, while the bargaining share in excess of the Hosios level compensates for the implied reduction in the workers'outside option.

Despite their negative e¤ects on aggregate welfare, we can characterize 1 See [START_REF] Hosios | On the E¢ ciency of Matching and Related Models of Search and Unemployment[END_REF].

a coalition in favor of ALMPs2 . These are favored by the least productive job seekers (or "short-term" unemployed") and the least productive workers.

The former gain directly from the subsidy, and the latter gain from an enhanced outside option in bargaining. On the other hand, more productive workers and job seekers lose from it. They are harmed due to the fall in the job …nding rate, which re ‡ects in particular the deterioration in average job seeker quality. Finally, the workers who do not search (or "long term unemployed") only bene…t if they are su¢ ciently close to the extensive margin of searching, that is, su¢ ciently productive. The least productive long-term unemployed are too far from the extensive margin of job search to bene…t from the policy, and su¤er from the …nancial burden of the search subsidy. Consequently, they oppose the policy. Note however that this analysis would be changed if ALMP were explicitly targeted at the least productive unemployed workers. Here, instead, by monitoring job search irrespective of productivity, the policy is implicity targeted at those workers whose productivity level is immediately below the critical search threshold. This paper is related to the recent literature on labor market policy analysis in the context of frictional models, following Mortensen and Pissarides (2014). This literature has analyzed to some details the e¤ect of unemployment bene…ts, often in the context of calibrated numerical analysis (see for example [START_REF] Cahuc | Should Unemployment Bene…ts Decrease with the Unemployment Spell ?[END_REF], [START_REF] Fredriksson | Optimal Unemployment Insurance in Search Equilibrium[END_REF], [START_REF] Mortensen | Taxes, Subsidies, and Equilibrium Labor Market Outcomes[END_REF], [START_REF] Lehmann | On the Optimality of Search Matching Equilibrium When Workers Are Risk Averse[END_REF], [START_REF] Krusell | Labour Market Matching with Precautionary Savings and Aggregate Fluctuations[END_REF], [START_REF] Michau | Optimal Labor Market Policy with Search Frictions and Risk-Averse Workers[END_REF]). The main novelty here is the focus on ALMP and the role played by the extensive margin of job search, which introduces a new externality (See [START_REF] Ortega | Pareto-Improving Immigration in an Economy with Equilibrium Unemployment[END_REF] for a related e¤ect when search is not directed and workers are heterogeneous). Also, the design of the model allows us to provide analytical results instead of relying on numerical simulations.

The paper is organized as follows. Section 2 spells out the basic framework, which is a standard matching model with heterogeneous worker productivity and a …xed search cost. Section 3 computes the equilibrium. Section 4 compares it with the utilitarian welfare optimum, and proves the …rst main result of the paper, i.e. that if the Hosios condition holds, the market outcome is associated with a suboptimally low job seeker quality and too much worker search, i.e. too small labor market tighter. Section 5 studies the e¤ect on welfare of active labor market policies and shows that the …rst best equilibrium can be replicated if the worker bargaining share and the search subsidy are set at their optimal levels, that are characterized by a bargaining share higher than the Hosios level and a search tax. Also, I characterize the e¤ect of ALMP on di¤erent categories of workers. There, the key result is that there exist two threshold levels for worker productivity (denoted by y), ỹe and ỹu ; such that ỹu > ỹe , such that the employed (resp. the unemployed job seekers) oppose an increase in the search subsidy if and only if y > ỹe (resp. ỹu > ỹe ).

The basic framework

Workers di¤er by their productivity y; and the population distribution of y is given by a density (y) and c.d.f. (y): At any point in time, unemployed workers may be searching or not searching -in the latter case their utility is equal to zero. We distinguish between u t ; the total number of unemployed workers, and u t ; those who are actively searching. The matching rate per unit of time is m( u t ; v t ) and the labor market tightness parameter is de…ned as = v= u: As usual, m() has constant returns to scale and the vacancy …lling rate is denoted by q( ) = m(1= ; 1); while the job …nding rate is q( ):

In order to be searching workers must incur a unit cost equal to d per unit of time. There is a constant job loss rate equal to s: The wage of a worker of type y is denoted by w(y): The value functions for being employed V e , unemployed V u , and the value of a job J depend on the worker's type and their valuation equations are, assuming steady state,

rV u (y) = d + q( )(V e (y) V u (y));
rV e (y) = w(y) + s(V u (y) V e (y)); rJ(y) = y w(y) sJ(y);

while wages are set by a standard Nash bargaining process with a fraction ' of the net surplus going to the worker. This implies the following equilibrium relationship:

V e (y) = V u (y) + ' 1 ' J(y):
Eliminating V e ; J; and w from these 4 equations, we can get V u (y) for a given and we get

rV u (y) = d(r + s) + ' q( )y r + s + ' q( ) : (1)
It is then easy to compute the value of being employed for a worker of type y :

rV e (y) = d(r(1 ') + s) + '( q( ) + r)y r + s + ' q( ) :

(2)

Finally the wage is

w(y) = '(r + s + q( ))y (1 ')(r + s)d r + s + ' q( ) :
We note that the search cost d brings wages down, by reducing the opportunity cost of work.

Equilibrium

To characterize the equilibrium, we need to derive the job creation condition.

We denote by y the average productivity of job applicants. The value of a worker with productivity y to the …rm is

J(y) = y w(y) r + s = (1 ')(y + d) r + s + ' q( ) :
Free entry of vacancies implies that EJ(y) = c=q( ); where the expectations are taken with respect to the pool of job applicants. Since J is linear in y; this is equivalent to J( y) = c=q( ); or equivalently

y + d = c(r + s) (1 ')q( ) + ' 1 ' c : (3)
This job creation locus de…nes an increasing relationship between y and : The tighter the labor market, the greater the …rms'search costs and the greater the average productivity of applicants must be to compensate.

Next, we need to know which workers search and which workers do not search. A worker of type y searches, in steady state, if and only if V u (y) > 0:

Using (1), we get that this is equivalent to

d(r + s) + ' q( )y > 0:
Therefore, there exists a critical productivity level y above which workers search, and

y = d(r + s) ' q( ) : (4)
In steady state, the average productivity of both job applicants and employed workers is then equal to

y = E(y j y > y ) = R +1 y y (y)dy 1 (y ) :
Clearly, d y=dy > 0: Therefore, (3) alternatively de…nes a positive relationship between y and : By contrast, (4) de…nes a decreasing relationship between and y : When the labor market is tighter, so is the job …nding rate which induces more unemployed workers to search. Accordingly the productivity threshold falls.

Equilibrium is then determined, as on Figure 1, by the intersection between the …rms'job creation condition JC, de…ned by (3), and the worker's search condition WS, de…ned by (4). This intersection de…nes the market outcome values of and y ; denoted by M and y M respectively.

Consider, for example, an increase in the workers'bargaining power ': It shifts both loci down (Figure 2). As a result, unambiguously falls but y may go up or down. Firms post fewer vacancies because they appropriate a smaller fraction of the surplus of the match. Workers search more for any because they appropriate a greater fraction of the surplus. But as falls, this per se discourages worker search, so that the overall e¤ect on y is ambiguous.

We can also note that in steady state, the unemployment rate is given by

u 1 = (y ) + (1 (y )) s s + q( ) = s + (y ) q( ) s + q( ) :
Consequently, if the net e¤ect of an increase in ' on y is negative, it may be that the unemployment rate is lower in the long run. Despite the fall in job …nding rates, expectations of appropriating a greater fraction of the surplus brings some of the "long-term unemployed" back into job search.

Social welfare

It is interesting to compare the preceding equilibrium with the social optimum. To be able to do this we need to formulate the social planner's problem adequately. There generally is an in…nite number of state variables, given by the unemployment rate of type y at date t; u(t; y): Total output at t must then be equal to

y t = Z +1 0 y(1 u(t; y)) (y)dy:
At each date the social planner chooses the vacancy rate v t ; or equivalently the degree of labor market tightness t ; as well as the minimum productivity level y t for workers to search. Therefore, the evolution equations of u(t; y) are given by _ u (t; y) = s(1 u(t; y)); y < y t ; _ u (t; y) = s(1 u(t; y)) t q( t )u(t; y); y > y t :

The social planner's objective function is given by max

Z +1 0 [ (c t + d) u t + y t ] e rt dt;
where u t ; the stock of unemployed workers actively searching, is given by

u t = Z +1 y u(t; y) (y)dy:
That is, the social planner maximizes the present discounted value of output net of …rms'(cv = c u) and workers'(d u) search costs.

The co-state variable associated with u(t; y) is denoted by [ (t; y)e rt (y)] :

The quantity (t; y) is interpreted as the marginal social value of an additional employed worker of type y: We can now write down the Hamiltonian:

H = [ (c t + d) u t + y t ] e rt e rt R y 0 (t; y)s(1 u(t; y)) (y)dy + R +1 y (t; y) (s(1 u(t; y)) t q( t )u(t; y)) (y)dy ! :
Next, we can write down the FOC:

@H @ t = 0 () c u t + (q( t ) + t q 0 ( t )) Z +1 y (t; y)u(t; y) (y)dy = 0; (5) @H @y t = 0 , c t + d (t; y ) t q( t ) = 0; (6)
@H @u(t; y) = @ @t (t; y) r (t; y) e rt (y)

, y + (r + s) (t; y) = @ @t (t; y); y < y y (c t + d) + (r + s + t q( t )) (t; y) = @ @t (t; y); y > y

Let us now focus on the steady state. We note that the marginal value of an employed worker of type y is, in steady state:

(t; y) = y r + s for y < y (t; y) = y + c + d r + s + q( )
for y > y :

Substituting into (6) allows to compute y ; and we get

y = (c + d) (r + s) q( ) : (8)
This condition de…nes the socially optimal search threshold for the workers. We can check that is continuous at y = y : Finally in steady state we have u(t; y) = s=(s+ q( )) = u for all y > y :

It follows that u = u(1 (y ))
and, substituting (7) into (5), we get

c(r + s 2 q 0 ( )) = (E(y j y > y ) + d)(q( ) + q 0 ( )): (9)
This condition de…nes the socially optimal job creation condition.

Comparing equilibrium and optimum.

To compare the equilibrium and the optimum, we need to confront ( 8)-( 9)

with ( 4)-( 3). Comparing the equilibrium and optimum job creation condition, i.e. ( 3) and ( 9), it is straightforward to check that the usual Hosios condition ' = q 0 ( )=q( ) must hold. However, this condition is no longer su¢ cient. For the two worker search condition to match, we would need in addition that

' = d c + d ;
which generally does not hold.

The term c in the denominator of the preceding formula captures the congestion externality exerted by an unemployed worker who decides to search. So far there is nothing new in this line of reasoning and it does not highlight why here (contrary to the usual case) the Hosios conditions fail to internalize the congestion externalities. The reason is that this negative externality is exerted upon an unemployed worker of average productivity, whereas the marginal unemployed worker only takes into account his own productivity level when considering the gains from search.

Consequently, even at the Hosios conditions, the degree to which the marginal worker internalizes the congestion problem is inadequate, because his productivity di¤ers from the productivity of the workers upon which the externality is exerted (a "quality e¤ect"). Since the marginal worker is less productive than the average, this e¤ect tends to lead to too much worker search. Therefore we expect the critical productivity level to be lower in the market equilibrium than in the centrally planned solution if the Hosios conditions hold -since the usual congestion externalities are internalized at the Hosios conditions, only the quality e¤ect dominates.

The following Proposition formally shows that, if the Hosios conditions hold, both labor market tightness and the critical productivity level (and hence the average productivity of the employed) are too low in the decentralized equilibrium compared to the social optimum.

Proposition 1 -Assume q( ) = and ' = : Then y M < y SP ;

M < SP :

Proof -Assume q( ) = and ' = : Clearly, then, in the ( ; y ) plane the decentralized worker search schedule WS de…ned by (4) and the socially optimal worker search schedule SSP de…ned by (8) intersect only once at

= ~ = d(1 ) c : (10)
Furthermore, computing the derivative of the RHS of (8) shows that this is the minimum point of SSP. It is straightforward to compute the value of y at this intersection point. It is given by

ỹ = (r + s) c 1 d (1 ) 1 : (11)
Next, we can show that the decentralized JC condition (3) coincides with the social one (8). Their common expression is 9) and (3) coincide. The social planner's optimal worker search condition is given by SSP (de…ned by ( 8)) and the corresponding market condition is WS (expressed by ( 4)). These two conditions intersect at a point ~ ; which furthermore is the minimum point of SSP. The JC condition is below the intersection point, so that it cuts WS before SSP, on the right of this minimum point.

y + d = c(r + s) 1 + 1 c : Substituting (
The LHS of ( 8) and ( 4) is the marginal bene…t -expressed in terms of the additional ‡ow of output generated by that worker when eventually employed -of putting an additional unemployed worker into active search.

The corresponding RHSs are the social and private marginal costs of doing so, respectively. The social planner considers the congestion cost imposed on the average job seeker, while at the Hosios conditions the marginal job seeker only internalizes the congestion costs imposed on marginal workers.

Welfare e¤ects of active labor market policies

Now we assume that the government pursues an active labor market policy, understood as a subsidy to job search which reduces the cost of search from d to d : I assume that somehow the government can enforce this policy, i.e. the subsidy is not paid to the workers who do not search. That is, the unemployed's search activity is "monitored" and is paid only upon proof of search activity. This makes the policy resemble more an active labor market policy than an unemployment bene…t system.

Furthermore, all unemployed job seekers are entitled to the subsidy, irrespective of their productivity or employment history. The analysis could deliver di¤erent results if, say, the subsidies were targeted to the low productivity workers 3 .

I assume the subsidy is …nanced by a lump-sum tax levied on all workers.

This tax does not introduce distortions in equilibrium determination, so the above computations for the equilibrium endogenous variables remain correct.

However the present discounted value of the tax now has to be deducted from the welfare of the employed and the unemployed, as de…ned in equations ( 3) and (2).

To compute this tax burden, we note that for a given search threshold y and a given distribution of unemployment rates by productivity levels u 0 (y);

the initial stock of unemployed workers actually searching is given by

u 0 = Z +1 y u 0 (y) (y)dy:
The total number of employed workers of types y > y at date t is given 3 In particular, in Saint-Paul (1998), active labor market policies harm the insiders, because they raise the search e¤ort of outsiders. But here the insiders would bene…t from the policy should they lose their job, which raises their bargaining power. Therefore, active policies may have very di¤erent e¤ects on the welfare of incumbent workers depending on how they are designed. by 1 (y ) u t : Consequently, the law of motion of u t is

d dt u t = s (1 (y ) u t ) q( ) u t :
The solution is

u t = ( u 0 u 1 ) e (s+ q( ))t + u 1 ;
where

u 1 = s (1 (y )) s + q( ) :
The tax cost of the subsidy at date t is

T t = u t ;
therefore the PDV of this tax is

Z +1 0 T t e rt dt = H;
where

H = u 0 r + s + q( ) + u 1 (s + q( )) r(r + s + q( )) = u 0 r + s + q( ) + s(1 (y )) r(r + s + q( ))
:

The utility function of an unemployed worker who is searching and has a productivity y can then be rewritten as

V u (y) = ( d)(r + s) + ' q( )y r(r + s + ' q( )) H: (12)
Similarly for the employed:

V e (y) = ( d)(r(1 ') + s) + '( q( ) + r)y r(r + s + ' q( )) H: (13)
We also need to discuss how a¤ects the job destruction and worker search conditions. For this it is enough to replace d by d in those conditions, and we get

y + d = c(r + s) (1 ')q( ) + ' 1 ' c ; (14) y = (d )(r + s) ' q( ) : (15)
We see that in the ( ; y ) plane an increase in shifts WS down and JC to the left. Labor market tightness unambiguously falls while the change in the average quality of workers is ambiguous.

The subsidy to job search raises the opportunity cost of work for those worker types who actively search. This increases wage pressure thus reducing pro…tability and the incentives to post vacancies. Therefore falls. Furthermore at the extensive margin, given ; more workers want to search. As such this e¤ect tends to further reduce because the additional workers are less productive than average, thus reducing again the value of posting vacancies. However the fall in per se tends to discourage job search, and if this fall is strong enough worker quality actually goes up, and so does long-term unemployment, despite the subsidy to job search.

The e¤ects of are qualitatively similar to those of ' : Both parameters shift the two loci in the same direction. A greater ' increases the worker's power in wage setting through the rent they can extract from the employer, while does it through their outside option in bargaining. A greater ' reduces y given because the prospects of greater rents induce more workers to search, while a greater does so through direct subsidization of search.

We are now in a position to discuss the e¤ects of ALMP on welfare.

We …rst start with social welfare and then proceed to discuss the welfare of di¤erent categories of workers.

Social welfare

While we already know that the Hosios conditions per se are insu¢ cient to restore e¢ ciency, we can analyze which combination of ' and delivers the …rst best. Admittedly this is a contrived exercise since presumably is a policy variable while ' is not. But ' can be targeted indirectly by the policymaker through regulations such as severance payments 4 .

We need to match the optimality condition with the equilibrium ones.

The equilibrium job creation condition ( 14) must coincide with the optimality condition (9) for y = y SP and = SP : Let = SP q 0 ( SP )=q( SP ): Then eliminating y = E(y j y > y SP ) between these two conditions and dropping the "SP " subscript we get the following:

= c(r + s + q( )) q( )(1 ) c(r + s + ' q( )) q( )(1 ') : (16)
This de…nes a decreasing, concave relationship, called OJC, in the ('; ) plane which goes through the point ( ; 0): This relationship depicts the combinations of ' and that make …rms internalize the congestion externality in job search. These are the combinations that deliver the correct social opportunity cost of labor. If for example the fraction of the surplus appropriated by the worker is greater than the Hosios level, then the cost of labor is too high and one has to tax search to reduce the worker's outside option in bargaining, thus bringing the cost of labor down back to the correct level from the social planner's perspective.

Similarly, we can match the equilibrium worker search condition (15) with the optimality one (8) and we get

= 'c + (1 ')d: (17)
This de…nes another decreasing, linear relationship between and ':

These are the combinations of ' and that deliver the socially optimal bene…t of search to the unemployed workers, re ‡ecting both the quality and congestion externalities. Since we know that the latter is internalized by 4 See for example the analysis in Saint-Paul, 2014. Of course, how allocative severance payments are depends on the degree of commitments of …rms and workers in bargaining. Under full commitment, a severance payment will be o¤set by a transfer upon recruitment, and will have no e¤ect on equilibrium job creation. However, under imperfect commitment, the severance payment will raise the worker's threat point in bargaining whenever wages are being renegotiated. workers at ' = ; only the (negative) quality externality remains, implying that along this locus < 0 at ' = : search must be taxed for workers to internalize the negative e¤ect of the marginal job seeker on the average quality of the pool 5 .

The joint determination of the optimal ' and is depicted on Figure 4. The OJC locus depicts the relationship ( 16), while OWS represents (17).

The optimal solution can then be characterized relative to the benchmark Hosios condition without ALMP as follows:

Proposition 2 -The optimal values of ' and are such that ' > and < 0: PROOF -Clearly, from the RHS of ( 16), we have that lim '!1 = 1 along OJC. We already know that the RHS of ( 16) is greater than that of (17) at ' = : Since the RHS of ( 17) remains …nite at ' = 1; by continuity there exists a value of ' 2 ( ; 1) ; ' ; at which OJC and OWS cross. Clearly, the corresponding is negative. Furthermore, the RHS of ( 16) is a concave function of '; while that of ( 17) is a¢ ne: Therefore, there is at most one other intersection point between OJC and OWS, and it must be such that ' < : For ' = 0; the RHS of ( 17) is equal to d: The RHS of ( 16) is equal to c(r + s + q( )) q( )(1 ) c(r + s) q( ) = c 1 r + s q( ) + :

5 Indeed this can be checked algebraically, by noting that

c SP + d SP q( SP ) (r + s) < c(r + s + SP q( SP )) q( SP )(1 ) d:
and then showing that this inequality is equivalent to

0 < 1 c d:
Finally, it is immediate that at ' = the RHS of ( 17) is negative. Now, from (9), and using (8), we have that

c 1 r + s q( ) + = d + y > d + y = d + (r + s) c q( ) + d r + s q( ) ;
from which it follows that c 1 r+s q( ) + > d: Therefore OJC is above OWS at ' = 0; and, by concavity, also over [0; ' ): Therefore, the only intersection between OJC and OWS is (' ; ):

QED

The optimal policy is to raise the worker's rent beyond the Hosios level while at the same time implementing a negative active labor market policy which taxes job search (we ignore feasibility constraints on such policies). This is just the opposite of what, say, an OECD report would recommend.

One way to interpret this result is as follows: starting from the Hosios condition value of '; search must be taxed because of the quality externality.

But taxing search reduces the cost of labor, leading to too high a vacancy level. To compensate for that, one must further raise the worker's bargaining power, which in turn must lead to a higher tax on job search 6 .

5.2 E¤ect on the welfare of di¤erent types of workers I now study which groups gain and lose from ALMPs. In the sequel I will assume that an increase in has a "normal" e¤ect on y ; i.e. that y falls 7 .

As a result it must be that dH=d > 0; both because y falls (a greater 6 Note however that this process converges to a consistent pair ('; ) only because (i) OJC becomes steeper relative to OWS as one moves to the right, and (ii) the opposite strategy of reducing ' and compensating by a raise in does not converge because OWS and OJC fail to cross on the left of ' = : This suggests that for some di¤erent version of the model one could have an optimum with > 0 and ' < instead. Intuitively, what is much more robust is the prediction that the optimal policy must satisfy (' ) < 0: Hence, if active labor market policy proves to be optimal, it must be accompanied with a reduction in the worker's bargaining power relative to the Hosios level.

7 Otherwise introducing a subsidy to job search would hardly qualify as "active" labor market policy.

proportion of the population is eligible for the subsidy) and falls (people who do search remain unemployed longer, so the subsidy has to be paid to them for a longer period).

Equations ( 12) and ( 13) imply that we can distinguish three kind of e¤ects of the subsidy on di¤erent categories of workers:

The negative direct tax e¤ects are given by d d ( H) and are the same for all workers, including those who do not search. These ones have a utility equal to H and they clearly are worse-o¤, unless the change in the subsidy makes them switch from non-search to search.

The direct positive e¤ect of the subsidy on the utility ‡ow while searching. This e¤ect is given by r + s r(r + s + ' q( ))

for the unemployed and by r(1 ') + s r(r + s + ' q( ))

for the employed. It is therefore stronger for the unemployed than for the employed, as their discounted expected time spent in unemployment is obviously larger. Furthermore this e¤ect does not depend on the worker's productivity y:

The indirect negative e¤ect on utility through the fall in : It is equal

to d d (q( ) + q 0 ( )) '(r + s)(y + d ) r(r + s + ' q( )) 2
for the unemployed and to d d (q( ) + q 0 ( )) '(r(1 ') + s)(y + d ) r(r + s + ' q( )) 2 for the employed. We notice that this negative e¤ect is also stronger for the unemployed, for whom the job …nding rate matters more than for the employed. Also, this e¤ect is stronger for more productive workers, because they appropriate part of the surplus of the match and therefore get higher wages, which makes them lose more from any reduction in job …nding rates. This suggests that, controlling for labor market status, more productive workers are more likely to oppose active labor market policies.

The preceding formulas allow us to …nd out who gains and who loses from an increase in among the employed and the unemployed. Consolidating all the terms spelled out above, we see that the marginal gains can be written as (r + s) (y) d d ( H) for the unemployed and (r(1 ') + s) (y) d d ( H) for the employed, where (y) = 1 r(r + s + ' q( )) + d d (q( ) + q 0 ( )) '(y + d ) r(r + s + ' q( )) 2 ; and 0 < 0: Therefore we see that an employed worker opposes the increase in i¤ his productivity level is greater than

ỹe = 1 ( 1 r(1 ') + s d d ( H)):
In particular, if ỹe < y ; all the employed opposed ALMPs.

As for the unemployed, their corresponding critical productivity level is

ỹu = 1 ( 1 r + s d d ( H)) > ỹe :
This inequality means that unemployed searchers are more in favor of active labor market policies than the employed.

However, the long-term unemployed, i.e. those such that y < y ; oppose it, except a tiny band of workers who are just below the critical search threshold and who switch their behavior because of the subsidy (but this band would not be tiny if we were considering a non in…nitesimal increment in ). Thus, somewhat paradoxically, here most of the long-term unemployed oppose ALMPs, however this is because here worker search only operates through the extensive margin. More generally, though, it makes sense to think that, to the extent that they have to contribute to their costs as taxpayers, the most disenfranchised long-term unemployed do not particularly support active labor market policies as it is unlikely to raise their own job …nding rate.

  This decision would reduce and to prevent it from falling, vacancies have to rise by an amount ; implying that an extra vacancy cost c must be spent. This extra cost is taken into account by the central planner but not by the individual worker. If the worker's appropriability level is equal to the ratio between his private search cost d and the social one c + d; then the congestion externality is internalized.

  Figure 3 illustrates Proposition 1. If ' = the Hosios conditions hold and the JC loci (9) and (3) coincide. The social planner's optimal worker search

  Figure 1 -Equilibrium determination

The political economy of unemployed bene…ts was analyzed in a pioneering paper byWright (1986). Active labor market policy was analyzed from a political economy perspective by[START_REF] Saint-Paul | A Framework for Analyzing the Political Sup[END_REF][START_REF] Ortega | Pareto-Improving Immigration in an Economy with Equilibrium Unemployment[END_REF].[START_REF] Boeri | The Political Economy of Flexicurity[END_REF] focus on the arbitrage between unemployment bene…ts and employment protection.


Jared Lolli

Alexandre Baksic

David Nagy

Vladimir E Manucharyan

Cristiano Ciuti

Ancillary qubit spectroscopy of cavity (circuit) QED vacua

Keywords: numbers: 42.50.Nn, 42.50.Pq, 05.30.Rt, 03.65.Yz

We investigate theoretically how the spectroscopy of an ancillary qubit can probe cavity (circuit) QED ground states containing photons. We consider three classes of systems (Dicke, Tavis-Cummings and Hopfield-like models), where non-trivial vacua are the result of ultrastrong coupling between N two-level systems and a single-mode bosonic field. An ancillary qubit detuned with respect to the boson frequency is shown to reveal distinct spectral signatures depending on the type of vacua. In particular, the Lamb shift of the ancilla is sensitive to both ground state photon population and correlations. Back-action of the ancilla on the cavity ground state is investigated, taking into account the dissipation via a consistent master equation for the ultrastrong coupling regime. The conditions for high-fidelity measurements are determined.

In recent years, cavity quantum electrodynamics (QED) has thrived thanks to the possibility of controlling light-matter interaction at the quantum level, which is relevant for the study of fundamental quantum phenomena, the generation of artificial quantum systems, and quantum information applications [START_REF] Haroche | Exploring the Quantum: Atoms, Cavities, and Photons[END_REF]. The field has more recently blossomed in solid-state systems, particularly in superconducting quantum circuit QED [START_REF] Wallraff | [END_REF]3] and semiconductor cavity QED [4]. In conventional cavity QED, photons are present only in the excited lightmatter states of the system and can escape the cavity due to a finite transparency of the mirrors. The situation changes drastically in the so-called ultrastrong lightmatter coupling regime [5][6][7][8][9][10], achieved when the lightmatter interaction rate is comparable or larger than the photon frequency. Indeed, it can become energetically favorable to have photons in the ground state. However, such ground state photons are bound to the cavity and cannot escape, since that would violate the energy conservation [11]. In the 'thermodynamic' limit where a large number N of two-level systems are (ultra)strongly coupled to a single bosonic mode, phase transitions can occur with non-trivial and degenerate vacua. The vacuum properties depend on the details of the light-matter coupling and on the Hamiltonian symmetries. These phase transitions are associated with a symmetry breaking: it is a discrete Z 2 symmetry for the phase transition [12][13][14][15] in the Dicke model [16] where the non-rotating wave terms of light-matter interaction are considered; it is a continuous U (1) symmetry in the case of the Tavis-Cummings model [17,18] where non-rotating wave terms are absent. Such symmetries can be controlled in models where the twolevel systems are coupled to both the quadratures of the bosonic field [19]. On the other hand, in Hamiltonians containing a strong enough quadratic renormalization of the cavity photon frequency (e.g., due to the squared electromagnetic vector potential term), the ground state is a two-mode squeezed vacuum, but no phase transition occurs [20]. This is the case for the Hopfield model [21], notably realized in semiconductor microcavities [5,[7][8][9]. The fundamental meaning and validity of cavity and circuit QED quantization procedures is critically at play in the ultrastrong coupling regime, since different forms of Hamiltonians lead to extremely different physical phenomena [22][23][24]. Protocols to detect the properties of cavity vacua are therefore of strong significance, not only for a study of intriguing ground states, but also as a sensitive test of fundamental microscopic theories.

In this context, a crucial question arises: how to detect ground state photon populations and correlations without destroying them? In principle bound photons in cavity (circuit) QED vacua can be released by a nonadiabatic, ultrafast modulation of the Hamiltonian parameters [5,[25][26][27][28], which can convert a ground state into an excited state. While non-adiabatic QED provides an interesting way of observing non-classical quantum vacuum radiation, finding a non-invasive and sensitive probe of the ground state properties remains highly desirable. A theoretical work [29] in circuit QED has suggested to study the coupling between a Dicke system and an additional superconducting qubit, showing that Dicke criticality can be observed via current transport measurements. However, in Ref. [29] the considered effective dispersive interaction between the cavity system and the auxiliary qubit depends only on the cavity photon population, and not on the intracavity light-matter correlations; moreover the dissipation has not been treated with a master equation suited for the ultrastrong coupling regime [30], which is essential to avoid artifacts such the instability of the ground state and the excitation of the system in the absence of driving [11,30].

In this Letter, we show that the spectroscopy of an ancillary qubit, moderately coupled to a cavity QED system, depends sensitively on the type of vacua. By driving this ancillary qubit and analyzing its frequency response, FIG. 1: A sketch of a cavity (circuit) QED system S consisting of a single-mode resonator coupled to N two-level (artificial) atoms. An ancillary qubit M is coupled to the resonator boson mode. The spectroscopy of the ancilla is used to probe quantum ground state properties of S.

we show that it is possible to have distinct signatures of the ground state photon populations and correlations without destroying them. We explore this protocol by considering three different classes of systems described respectively by the Dicke, Tavis-Cummings and Hopfield models, each one having a ground state with different properties. We show numerically and analytically that the Lamb shift of the ancillary qubit transition is very sensitive both on the photon populations and correlations of exotic vacua. We explore the back-action of the ancillary qubit on the cavity ground state and determine the key physical quantities affecting the fidelity of the measurement, including consistently the dissipation effects in the ultrastrong coupling regime.

As sketched in Fig. 1, we will consider an ancillary qubit M coupled to the bosonic mode of a cavity (circuit) QED system S. In particular, we will deal with the following time dependent Hamiltonian ( = 1),

H(t) = H S + ω M 2 σ (M ) z +g M a † + a σ (M ) x +Ω p cos(ω p t)σ (M ) x , (1)
where H S is the system Hamiltonian, g M is the coupling between the measurement qubit and the boson field, whose boson annihilation operator is a. The σ (M ) i Pauli operators act on the Hilbert space of the qubit M , whose transition frequency is ω M , while Ω p is the amplitude of the driving field with frequency ω p acting on M .

In the following, H S will be one of the three Hamiltonians, describing respectively the Dicke, Tavis-Cummings and Hopfield-like models ( = 1):

H Dicke = ω c a † a + ω 0 J z + λ √ N a † + a (J + + J -) ,
H T C = ω c a † a + ω 0 J z + λ √ N a † J -+ aJ + , H Hopf ield = H Dicke + D a † + a 2 ,
where ω c is the frequency of the bosonic mode, ω 0 is the transition frequency of each of the N identical two-level atoms, λ is the collective coupling and D = λ 2 /ω 0 is the strength of the boson renormalization term in the Hop- field model. The J i are the angular momentum operators representing the collective pseudo-spin associated to the N two-level systems, namely

J z = 1 2 N i=1 σ (i) z , J ± = N i=1 σ (i)
± , where the Pauli matrices here refer to each two-level system.

We start by considering the energy levels of H S+M , describing the system S coupled to the measurement qubit M , namely:

H S+M = H S + ω M 2 σ (M ) z + g M a † + a σ (M )
x .

(

) 2
The eigenstates and their energies are defined by H S+M |l = ǫ l |l . System S will be of the Dicke, Tavis-Cummings or Hopfield type, as shown in Fig. 2. We consider a qubit transition frequency ω M detuned with respect to the boson frequency ω c = ω 0 . For g M = 0, the driving field term, proportional to the operator σ The top, middle and bottom panels of Fig. 2 are respectively for the Dicke, Tavis-Cummings and Hopfield models. For λ = 0 the system S is like a bare cavity, so the spectral renormalization is the standard Lamb shift [31] of the qubit due to the coupling to the normal vacuum in the cavity. By increasing λ the vacuum is modified and the Lamb-shift changes. It is apparent that the behavior of the qubit shift is qualitatively different in the three cases. For the Dicke model (top panels), the Lamb shift increases strongly with λ and becomes much bigger than in the case of the bare cavity (λ = 0). In the Tavis-Cummings case (middle panels), the Lamb shift increases in a step-like way as a function of λ. In the Hopfield model (bottom panels), instead the Lamb shift decreases with increasing value of λ.

By generalizing the approach in Ref. [33], we have derived an analytical expression at the second order in g M for the measurement qubit Lamb shift, namely

δω (S) M ≃ g 2 M 1 ω M -ω c + 1 ω M + ω c (a + a † ) 2 + g 2 M 1 (ω M -ω c ) 2 - 1 (ω M + ω c ) 2 V (S) , (3)
where

V (Dicke) = λ √ N (a + a † )J x , V (T C) = λ √ N (aJ + + a † J -) and V (Hopf ield) = V (Dicke) + 2 λ 2 ω0 (a + a † ) 2 .
Here the expectation values are calculated on the ground state |G S of the target system S. Importantly, the shift not only depends on the ground state photon population a † a , but also on the anomalous expectation value a †2 + a 2 and on the correlation between photon field and the N two-level systems. The red-dashed lines in the right panels of Fig. 2 depict the shift predicted by Eq. ( 3). The agreement between the numerical diagonal-ization results and the analytical formula is excellent in the considered range of values for λ/ω c , except for points where there are avoided crossings with other levels.

In Fig. 3, we present the behavior of the qubit spectral shift as a function of N . With increasing value of N , a critical point emerges for the Dicke Hamiltonian (left panels), but not for the Hopfield case (right panels). The behavior of the qubit Lamb shift, already completely different for small values of N , becomes strikingly dissimilar. Already for N = 30, the shift has a considerable jump around the critical coupling. The bottom panels shows the ground state fidelity

F G = T r S,M (|G S+M G S+M |(|G S G S | ⊗ 1(M) )
), quantifying the change of the cavity system ground state in presence of the ancilla qubit. In the considered conditions, F G can be close to 1. However, for large values of λ/ω c the fidelity strongly decreases in the Dicke case above the critical coupling, while it stays close to 1 for Hopfield. Since for g M → 0 the fidelity tends to 1 and the qubit shift is proportional to g 2 M , a trade-off between fidelity and size of the qubit shift can be found depending on the degree of level broadening.

In order to include dissipation consistently with the ultrastrong coupling regime, we need to consider a master equation for the density matrix where the quantum jumps occur between the actual eigenstates of the Hamiltonian H S+M [30,32]. We consider three decay channels, associated to the bosonic mode, the N two-level systems and the measurement qubit, with dissipation rates γ c , γ 0 and γ M respectively (see Fig. 1). Namely:

ρ = -i[H(t), ρ]+ γ c 2 D[U [a † +a]]+ γ 0 2 D[U [J x ]]+ γ M 2 D[U [σ (M ) x ]] (4) where D[ Â] = 2 Âρ Â † -ρ Â † Â -Â † Âρ and U [ Â] = ll ′ Θ(ǫ l ′ -ǫ l ) l| Â|l ′ |l l ′ |
defines the jump operators taking as argument the system operator involved in the coupling to the reservoir (Θ(ω < 0) = 0 and Θ(ω > 0) = 1). Here we are considering reservoirs at zero temperature (they can only absorb energy from the system S + M ). Note that if one uses the standard Lindblad equation with bare excitation operators as in Ref. [29], the ground state of the whole system H S+M is unstable and the reservoir excites the system even at zero temperature.

We can now apply the master equation in Eq. ( 4) to describe the spectroscopy when qubit M is driven as described by Eq. ( 1). We have determined the steadystate density matrix ρS+M and consequently the re- duced density matrix of qubit M and system S , namely ρM = T r S (ρ S+M ) and ρS = T r M (ρ S+M ). In Fig. 4, we show results for the qubit excited state population n

(M ) ↑ = T r S,M (ρ S+M (1 + σ(M) z
)/2) versus the driving frequency ω p and the collective atom-photon coupling λ. The ancilla excited state population spectrum shows a resonant peak that provides direct access to the vacuumdependent qubit Lamb shift discussed so far and well described by the formula in Eq. ( 3). Within our framework, we can evaluate the degree of back-action on the system S. In particular we can calculate the measurement fidelity F = T r S (ρ S |G S G S |), depending on the coupling between S and M , the driving of the qubit and the dissipation rates. F = 1 means that the cavity ground state is unaffected by the overall measurement process. In Fig. 5, we show F versus ω p for different values of λ and of the dissipation rates. The moderate dip at the resonance frequency is due to creation of real excitations in the system S via the driving of the qubit M . When the driving amplitude Ω p → 0, the dip disappears (not shown). Out of resonance, F → F G , the fidelity depending only on the level mixing between qubit M and system S (see Fig. 3), quantified by the ground state fidelity F G . Concerning the dissipation, our results show that when the cavity system S dissipation rates γ c , γ 0 are much smaller than the ancilla qubit dissipation rate γ M , then the most pronounced fidelity dip is obtained (black solidlines in Fig. 5 are for vanishing dissipation in the system S). Indeed, in such conditions a significant steady-state population of excited states can be created in S due to the low dissipation rates, implying that the ancilla qubit cannot 'read' faithfully the ground state of the system. Instead in the opposite limit, the fidelity dip disappears (F(ω dip ) → F G ) as excited state populations in S can be dissipated efficiently.

In conclusion, we have shown theoretically that the spectroscopy of an ancillary qubit coupled to a cavity (circuit) QED system is a very sensitive probe of its ground state photon properties. The spectral Lamb shift of the ancillary qubit transition is vacuum-dependent, namely depending on the ground state photon populations and correlations. The Lamb shift behaves qualitatively in a different way for systems described by the Dicke, Tavis-Cummings and Hopfield models, whose exotic vacua are qualitatively different. By a consistent solution of the master equation to include dissipation in the ultrastrong coupling regime, we have studied the measurement fidelity by accounting for level-mixing between system and measurement qubit, driving and dissipation. The present work demonstrates that ancillary qubit spectroscopy of cavity QED systems is a promising tool to study non-destructively the rich physics of QED vacua in the ultrastrong light-matter coupling regime.

C. C. acknowledges partial support from Institut Universitaire de France and from ERC Consolidator grant 'CORPHO'.

FIG. 2 :

 2 FIG.2:(Color online) Left panels: excitation energies for the three considered systems S versus the coupling λ between the boson field and the N atoms. Right panels: the Lamb shift of the ancillary qubit transition. The red dashed lines are calculated via Eq. (3). Top panels: Dicke system with N = 3, ωc = ω0, ωM = 2.75ωc, gM = 0.1ωc. Middle panels: Tavis-Cummings system with N = 3, ωc = ω0, ωM = 2.75ωc, gM = 0.1ωc. Bottom panels: an Hopfield system with N = 3, ωc = ω0, ωM = 6.75ωc, gM = 0.1ωc, D = λ 2 /ω0.

FIG. 3 :

 3 FIG. 3: Top panels: Lamb shift of the ancillary qubit (black dots) versus λ for N = 1, 3, 10 and 30. Red-dashed lines are obtained via Eq. (3). Bottom panel: ground state fidelity FG. Left panels: Dicke model with ωc = ω0, ωM = 2.75ωc, gM = 0.1ωc. Right panels: Hopfield model with ωc = ω0, ωM = 6.75ωc, gM = 0.1ωc, D = λ 2 /ω0.

  from the ground state |G S ⊗ | ↓ to the excited state |G S ⊗ | ↑ , being |G S the ground state of S and | ↓ (| ↑ ) the ground (excited) state of the qubit M . For finite g M , the coupling between S and M creates a mixing between states of the form |Ψ S ⊗ |ψ M and the driving induces a transition from the ground state |G S+M to excited states of H S+M . Therefore, in the spectroscopy the relevant excited states |l are those having the largest values of | G S+M |σ (M ) x |l | 2 . The color scale of the levels in Fig. 2 is proportional to such matrix element. The results show that, due to the off-resonant coupling, there is only one dominant spectroscopicallyactive level (black thick solid line), which has a strong overlap with the state |G S ⊗ | ↑ . The right panels shows the Lamb shift of the qubit transition frequency.

12 FIG. 4 :

 124 FIG. 4: Excited state population of the ancilla qubit M versus the coherent drive frequency ωp for different values of collective coupling λ. Left panel: Dicke model with Ωp = 0.5γM . Right panel: Tavis-Cummings model with Ωp = 0.2γM . Dissipation parameters: γM = γc = γ0 = 0.01ωc. The other parameters are as in Fig. 2. The white line corresponds to the analytical curve in Eq. (3).

FidelityFIG. 5 :

 5 FIG. 5: Measurement fidelity F (see definition in the text) for different values of λ for the Dicke model and for different dissipation rates γc = γ0 = ηγM and γM = 0.01ωc. Solid line: η = 0. Dashed line: η = 1. Dot-dashed line: η = 10. Other parameters as in Fig. 4.