polystyrene microspheres of diameter 1 m ( g. kisker - products for biotechnology ) are diluted in deionized and filtered water to a final concentration of a few spheres per millilitre . the spheres are inserted into a custom - made electrophoretic fluid chamber with two electrodes . a gaussian white noise signal is generated with an independent generator ( tabor electronics , ww1071 ) . the external noise is modulated by a custom - made voltage multiplier ( 100 khz bandwidth ) with a signal ( vt ) generated by a dual generator ( tabor electronics , dual channel ww5062 ) . the output signal of the multiplier is amplified 1,000 times with a high - voltage power amplifier ( trek , 623b ) before being applied to the electrodes . the optical potential is generated by a 980 nm laser beam which is inserted through an oil immersion objective ( nikon , cfi pl fl  100 na 1.30 ) into the fluid chamber . the detection of the motion of the particle is achieved by an additional 532 nm laser beam that is passed through the trapping objective . the forward scattered light is collected by an additional microscope objective (  10 , na = 0.10 ) , and its backfocal - plane field distribution is analysed by a quadrant position detector ( new focus 2911 ) at an acquisition rate of 2 khz . a laser controller ( arroyo instruments 4210 ) allows the management of the optical power at a maximum rate of 250 khz using an external voltage vk . the trap stiffness depends linearly on the optical power and can be controlled at the same rate as it . the signal sent to the laser controller ( vk ) is also generated by the dual generator , and is hence synchronized with vt . the electronic control of the protocol allows us to implement it at different cycle times without loss of resolution , ranging from  = 10 ms to  = 200 ms , during exp = 50 s. for simplicity , we impose a time - symmetric protocol for the stiffness , { (t)}t=0 with (t ) = (  t ) . the stiffness increases quadratically with time from t = 0 to t = /2 and decreases at the same rate from /2 to . we fix the minimum and maximum values of the stiffness , i = (0 ) = ( 2.0  0.2 ) pn m and iii = (/2 ) = ( 20.0  0.2 ) pn m respectively . for convenience , we define ii = (/4 ) = ( 6.5  0.2 ) pn m . the geometry of the carnot cycle imposes the value of iv = ( iii/ii)i = ( * ) = ( 6.2  0.2 ) pn m , where  * yields *=(/2)(iii/ii)1/(iii/i)10.76. the temperature of the particle remains constant at the isothermal steps : ti = tii = tc = 300 k during t  [ 0 , /2 ] and tiii = tiv = th = 525 k during t  [ /2, * ] . along the adiabatic steps , the temperature changes smoothly while t(t)/(t ) remains constant to ensure that the total shannon entropy of the system is conserved14,33 . for each cycle of duration  , we sample the position of the bead with respect to the centre along the x axis set by the direction of the external field . in all cases , we measure trajectories of the position { xt}t=0 with sampling rate 2 khz ( t = 5 ms ) along exp = 50 s. the stochastic work exerted to the particle in the interval of time [ t , t + t ] in a single realization is given by ( 2)dwt = f(xt , t)dt = f(xt , t)+f(xt+t , t+t)2[t+tt ] where  denotes the stratonovich product ( ref . 6 ) and f(xt , t)=u(xt , t)/t = xt2/2 is the generalized force conjugated to the control parameter . the stochastic heat transferred in [ t , t + t ] from the effective thermal bath to the particle is calculated using the first law of thermodynamics : ( 3)dqt = dhtdwt = dut+dekin , tdwt where dut=(1/2)(t+txt+t2txt2 ) is the change in potential energy , dekin , t=(m/2)(vt+t2vt2 ) is the kinetic energy change , and dwt is given by equation ( 2 ) . equation ( 3 ) equals sekimoto s celebrated expression for microscopic heat6 . both work and heat along the elementary processes in the cycle are calculated by summing the contributions of equations ( 2 ) and ( 3 ) from the beginning to the end of the process . the kinetic energy of the particle ( of mass m and friction coefficient 
  ) at time t , vt , is 
 calculated from the time - averaged velocity at time t , vt=(xt+txt)/t : ( 4)vt2=tvt2 where t is a 
 correction factor that depends on the acquisition frequency f = 
 1/t and on the physical parameters of the system at 
 time t : ( 5)t=12f2[1f02+efp2ff1(ef1/ffp+2f1ef1/ffp2f1)]1 where fp=/m , f=t/2,f0=fpf  and  f1=fp2/4f02 ( refs 
 13,14 ) . the shannon entropy of the particle at time t , st , is measured as the sum of the positional and kinetic entropy , st = sx , t + sv , t . here , sx , t=dxt(xt , t)ln(xt , t ) , where (xt , t ) is the pdf to observe trajectories that pass through xt at time t along the experiment . the distribution (xt , t ) is estimated from the histogram of xt using a regular binning of 25 bins in the interval [ 3(xt ) , 3(xt ) ] , with (xt ) being the standard deviation of the position at time t. the same procedure is applied to determine sv , t=kdvt(vt , t)ln(vt , t ) ln p(vt , t ) , binning in the interval [ 3(vt ) , 3(vt ) ] in this case , with vt being determined using equations ( 4 ) and ( 5 ) . for a given cycle of duration  , we first calculate the distribution of (i ) for i = [ 1,2,3,  ,400 ] cycles . for each value of i , the pdf ( ) is estimated as follows : we first compute the values of (i ) ( equation ( 1 ) in the main text ) along the experiment as the ratio of the work summed over i consecutive cycles over the heat summed over the same i cycles . 3 are calculated using a kernel density routine in matlab r2013a by partitioning the data in regular bins ranging from 5c to 5c of width 0.02c . the traditional definition of efficiency for a heat engine operating between two thermal baths is the ratio between work extracted in the cycle and heat absorbed during the hot isothermal , 1,(i)=w(i)q3,(i ) if heat is exchanged during other steps different from isothermals , even if only because of fluctuations , it may be relevant to reflect this in the efficiency . thus , these two other definitions for the efficiency are possible : 2,(i)=w(i)q2,(i)+q3,(i),3,(i)=w(i)q2,(i)+q3,(i)+q4,(i ) for example , 2,(i ) is the type of efficiency usually considered when the heat in step 2 ( the step just before the hot isothermal expansion ) is exchanged with the hot bath , as in the microscopic stirling motor3 ( with step 2 corresponding to the isochoric heating in this case ) . the corresponding long - term efficiencies are obtained with the limits i   and    : 1=wq3,2=wq2,+q3,3=wq2,+q3,+q4, where w=limi(w(i)/i ) , and the same for the heat . theoretically , these three definitions of long - term efficiency converge to carnot efficiency , as heat exchanges in the adiabatic steps vanish on average . however , the experimental long - term efficiencies do not coincide for large  ( data not shown ) . this can be understood by noting that 3 is the quantity with the smallest fluctuations around c of the three in this limit . thus , when averaging over a finite number of cycles this approaches carnot efficiency faster . we can analyse the fluctuations of 3 in the quasistatic limit by noting that the work is delta distributed . then , the denominator in 3 can be expressed using the first law applied to the process 2  3  4 , as q2+q3+q4=hihiiw+w1 where w is the total work in the cycle and w1 is work in the cold isothermal , both deterministic quantities . here i and ii refer to the initial and final state of the system during the subprocess 1 . this yields 3=ww+w1+hihii where hiihi is the total energy difference between the initial and final state in the cold isothermal and is the only fluctuating quantity in the expression . the quasistatic averages of the work in the cold isothermal and in the cycle are equal to w = k2(tcth)logiiiw1=ktc2logiii and then 3=c1 + 2(hihii)kthlogiii the distribution of the internal energy change is given by (hihii)=cec|hihii| with c = 1/ktc . from this , we can compute the variance of hi  hii , which reads (hihii)=2ktc this gives a variance for 3 of 3=c|2kthlogiii|(hihii ) and for the relative fluctuation 3c=|22kthlogiii|ktc equivalently , the variance of 1 in the quasistatic limit reads 1c=|22kthlogiii|kth fluctuations on 1 are a factor th / tc stronger than those of 3 and values for 1 computed from averages of a small finite number of cycles will in general be less reliable than those of 3 . when considering the intermediate case of 2 , the variance can be shown to be equal to 2c=|22kthlogiii|k2th2+tc2 our theoretical results predict that the width of the long - term efficiencies satisfy 3  2  1 , which indicates than when estimating efficiencies 3 is expected to have the least experimental error associated with fluctuations . polystyrene microspheres of diameter 1 m ( g. kisker - products for biotechnology ) are diluted in deionized and filtered water to a final concentration of a few spheres per millilitre . the spheres are inserted into a custom - made electrophoretic fluid chamber with two electrodes . a gaussian white noise signal is generated with an independent generator ( tabor electronics , ww1071 ) . the external noise is modulated by a custom - made voltage multiplier ( 100 khz bandwidth ) with a signal ( vt ) generated by a dual generator ( tabor electronics , dual channel ww5062 ) . the output signal of the multiplier is amplified 1,000 times with a high - voltage power amplifier ( trek , 623b ) before being applied to the electrodes . the optical potential is generated by a 980 nm laser beam which is inserted through an oil immersion objective ( nikon , cfi pl fl  100 na 1.30 ) into the fluid chamber . the detection of the motion of the particle is achieved by an additional 532 nm laser beam that is passed through the trapping objective . the forward scattered light is collected by an additional microscope objective (  10 , na = 0.10 ) , and its backfocal - plane field distribution is analysed by a quadrant position detector ( new focus 2911 ) at an acquisition rate of 2 khz . a laser controller ( arroyo instruments 4210 ) allows the management of the optical power at a maximum rate of 250 khz using an external voltage vk . the trap stiffness depends linearly on the optical power and can be controlled at the same rate as it . the signal sent to the laser controller ( vk ) is also generated by the dual generator , and is hence synchronized with vt . the electronic control of the protocol allows us to implement it at different cycle times without loss of resolution , ranging from  = 10 ms to  = 200 ms , during exp = 50 s. for simplicity , we impose a time - symmetric protocol for the stiffness , { (t)}t=0 with (t ) = (  t ) . the stiffness increases quadratically with time from t = 0 to t = /2 and decreases at the same rate from /2 to . we fix the minimum and maximum values of the stiffness , i = (0 ) = ( 2.0  0.2 ) pn m and iii = (/2 ) = ( 20.0  0.2 ) pn m respectively . for convenience , we define ii = (/4 ) = ( 6.5  0.2 ) pn m . the geometry of the carnot cycle imposes the value of iv = ( iii/ii)i = ( * ) = ( 6.2  0.2 ) pn m , where  * yields *=(/2)(iii/ii)1/(iii/i)10.76. the temperature of the particle remains constant at the isothermal steps : ti = tii = tc = 300 k during t  [ 0 , /2 ] and tiii = tiv = th = 525 k during t  [ /2, * ] . along the adiabatic steps , the temperature changes smoothly while t(t)/(t ) remains constant to ensure that the total shannon entropy of the system is conserved14,33 . for each cycle of duration  , we sample the position of the bead with respect to the centre along the x axis set by the direction of the external field . in all cases , we measure trajectories of the position { xt}t=0 with sampling rate 2 khz ( t = 5 ms ) along exp = 50 s. the stochastic work exerted to the particle in the interval of time [ t , t + t ] in a single realization is given by ( 2)dwt = f(xt , t)dt = f(xt , t)+f(xt+t , t+t)2[t+tt ] where  denotes the stratonovich product ( ref . 6 ) and f(xt , t)=u(xt , t)/t = xt2/2 is the generalized force conjugated to the control parameter . the stochastic heat transferred in [ t , t + t ] from the effective thermal bath to the particle is calculated using the first law of thermodynamics : ( 3)dqt = dhtdwt = dut+dekin , tdwt where dut=(1/2)(t+txt+t2txt2 ) is the change in potential energy , dekin , t=(m/2)(vt+t2vt2 ) is the kinetic energy change , and dwt is given by equation ( 2 ) . equation ( 3 ) equals sekimoto s celebrated expression for microscopic heat6 . both work and heat along the elementary processes in the cycle are calculated by summing the contributions of equations ( 2 ) and ( 3 ) from the beginning to the end of the process . the kinetic energy of the particle ( of mass m and friction coefficient 
  ) at time t , vt , is 
 calculated from the time - averaged velocity at time t , vt=(xt+txt)/t : ( 4)vt2=tvt2 where t is a 
 correction factor that depends on the acquisition frequency f = 
 1/t and on the physical parameters of the system at 
 time t : ( 5)t=12f2[1f02+efp2ff1(ef1/ffp+2f1ef1/ffp2f1)]1 where fp=/m , f=t/2,f0=fpf  and  f1=fp2/4f02 ( refs 
 13,14 ) . the shannon entropy of the particle at time t , st , is measured as the sum of the positional and kinetic entropy , st = sx , t + sv , t . here , t ) , where (xt , t ) is the pdf to observe trajectories that pass through xt at time t along the experiment . the distribution (xt , t ) is estimated from the histogram of xt using a regular binning of 25 bins in the interval [ 3(xt ) , 3(xt ) ] , with (xt ) being the standard deviation of the position at time t. the same procedure is applied to determine sv , t=kdvt(vt , t)ln(vt , t ) ln p(vt , t ) , binning in the interval [ 3(vt ) , 3(vt ) ] in this case , with vt being determined using equations ( 4 ) and ( 5 ) . for a given cycle of duration  , we first calculate the distribution of (i ) for i = [ 1,2,3,  ,400 ] cycles . for each value of i , the pdf ( ) is estimated as follows : we first compute the values of (i ) ( equation ( 1 ) in the main text ) along the experiment as the ratio of the work summed over i consecutive cycles over the heat summed over the same i cycles . 3 are calculated using a kernel density routine in matlab r2013a by partitioning the data in regular bins ranging from 5c to 5c of width 0.02c . the traditional definition of efficiency for a heat engine operating between two thermal baths is the ratio between work extracted in the cycle and heat absorbed during the hot isothermal , 1,(i)=w(i)q3,(i ) if heat is exchanged during other steps different from isothermals , even if only because of fluctuations , it may be relevant to reflect this in the efficiency . thus , these two other definitions for the efficiency are possible : 2,(i)=w(i)q2,(i)+q3,(i),3,(i)=w(i)q2,(i)+q3,(i)+q4,(i ) for example , 2,(i ) is the type of efficiency usually considered when the heat in step 2 ( the step just before the hot isothermal expansion ) is exchanged with the hot bath , as in the microscopic stirling motor3 ( with step 2 corresponding to the isochoric heating in this case ) . the corresponding long - term efficiencies are obtained with the limits i   and    : 1=wq3,2=wq2,+q3,3=wq2,+q3,+q4, where w=limi(w(i)/i ) , and the same for the heat . theoretically , these three definitions of long - term efficiency converge to carnot efficiency , as heat exchanges in the adiabatic steps vanish on average . however , the experimental long - term efficiencies do not coincide for large  ( data not shown ) . this can be understood by noting that 3 is the quantity with the smallest fluctuations around c of the three in this limit . thus , when averaging over a finite number of cycles this approaches carnot efficiency faster . we can analyse the fluctuations of 3 in the quasistatic limit by noting that the work is delta distributed . then , the denominator in 3 can be expressed using the first law applied to the process 2  3  4 , as q2+q3+q4=hihiiw+w1 where w is the total work in the cycle and w1 is work in the cold isothermal , both deterministic quantities . here i and ii refer to the initial and final state of the system during the subprocess 1 . this yields 3=ww+w1+hihii where hiihi is the total energy difference between the initial and final state in the cold isothermal and is the only fluctuating quantity in the expression . the quasistatic averages of the work in the cold isothermal and in the cycle are equal to w = k2(tcth)logiiiw1=ktc2logiii and then 3=c1 + 2(hihii)kthlogiii the distribution of the internal energy change is given by (hihii)=cec|hihii| with c = 1/ktc . from this , we can compute the variance of hi  hii , which reads (hihii)=2ktc this gives a variance for 3 of 3=c|2kthlogiii|(hihii ) and for the relative fluctuation 3c=|22kthlogiii|ktc equivalently , the variance of 1 in the quasistatic limit reads 1c=|22kthlogiii|kth fluctuations on 1 are a factor th / tc stronger than those of 3 and values for 1 computed from averages of a small finite number of cycles will in general be less reliable than those of 3 . when considering the intermediate case of 2 , the variance can be shown to be equal to 2c=|22kthlogiii|k2th2+tc2 our theoretical results predict that the width of the long - term efficiencies satisfy 3  2  1 , which indicates than when estimating efficiencies 3 is expected to have the least experimental error associated with fluctuations .
the carnot cycle imposes a fundamental upper limit to the efficiency of a macroscopic motor operating between two thermal baths1 . 
 however , this bound needs to be reinterpreted at microscopic scales , where molecular bio - motors2 and some artificial micro - engines35 operate . 
 as described by stochastic thermodynamics6,7 , energy transfers in microscopic systems are random and thermal fluctuations induce transient decreases of entropy , allowing for possible violations of the carnot limit8 . here we report an experimental realization of a carnot engine with a single optically trapped brownian particle as the working substance . 
 we present an exhaustive study of the energetics of the engine and analyse the fluctuations of the finite - time efficiency , showing that the carnot bound can be surpassed for a small number of non - equilibrium cycles . as its macroscopic counterpart , the energetics of our carnot device exhibits basic properties that one would expect to observe in any microscopic energy transducer operating with baths at different temperatures911 . 
 our results characterize the sources of irreversibility in the engine and the statistical properties of the efficiency  an insight that could inspire new strategies in the design of efficient nano - motors .