M ICFP - Quantum Physics Hélène Perrin - Julien Laurat Atoms and Photons Quantum opto-mechanical devices Answers Classical field in a Fabry-Perot cavity Fixed mirrors x x n (ω ) x, fixed x' Figure : Fixed M mirror The origin of x axis is at the position of mirror M ω is defined as x n (ω ), ie the resonant frequency for a cavity length x n (ω ) On resonance, the cavity length is a multiple of λ/: nλ n / Using the relation ω n kc πc/λ n, we deduce that the eigenfrequencies ω n of the intra-cavity field read cavity length at rest without light x x' ω n n πc x resonant length for ω ω x n (ω) x x n (ω) + x They are spaced by the free spectral range ω πc/ At first order in γ: t ( r ) / [ ( γ) ] / (γ γ ) / γ 3 E cav + is the sum of all fields propagating to the right inside the cavity The initial transmission through the first mirror M gives a factor t Each successive propagation to the right mirror M and back, with % reflection, gives a phase shift k, and each successive reflection at M gives a factor r We have thus: ( E cav + te in + tre ik E in + t re ) ikl Ein + t re ik E in The laser frequency is close to a resonance at frequency ω n πc/ This means that the detuning is small as compared to the free spectral range: ω Let us write explicitly the wave vector k: k ω c ω c + c n π + c The exponential term thus simplifies in e i /c We remark that the exponent also reads iπ / ω and is small We can expand it and write re ik re i c r ir c γ i c
We neglect the difference between r and in the second term as / ω is already small The relation between the fields thus reads: E cav + t γ i c E in itc/() + i γc E in it/τ + i Γ E in () where we have defined τ /c π/ ω, the total round-trip time, and Γ γc/ t /τ, the inverse photon lifetime in the cavity Taking the modulus square, we can now write I + cav E+ cav E in t /τ + Γ 4 4t /(τ Γ ) + 4 /Γ 4/t + 4 /Γ The field propagating along x results in a standing wave in the cavity, with an average intensity I cav I + cav By identification, we find the relation I cav A + 4 /Γ, () with A 8/t 4/γ A is related to the cavity finesse F ω/γ π/γ through A 4F/π 4 Each photon reflected onto the mirror M has its momentum reversed, from + hk to hk The momentum change is thus hk Conversely, the mirror undergoes a momentum kick + hk at each reflection This phenomenon happens at a rate given by the number of photons impinging on the mirror per second, which is also the ratio between x n (ωthe ) power inside the cavity x associated, fixed with x' the light propagating to the right P cav + SI cav + SI cav / and the energy per photon hω We can finally write: x Moving mirror F hk P cav + hω k SI cav ω SI cav (3) c x n (ω ) x cavity length at rest without light resonant length for ω ω x n (ω) x x + x x' Figure : Moving M mirror is by definition the cavity length for a mirror at its equilibrium position, in the absence of light ω is defined as earlier, such that x n (ω ) At an driving frequency ω, the resonant length is x n (ω) The position of mirror M is x with respect to M, and x with respect to The laser frequency being given at ω, the mirror position where the cavity is resonant is: x x n (ω) n λ/ n πc/ω ω /ω We will still use ω n πc/, but the resonance frequency ω res (x) for a given x now differs from ω and depends on x We have: ω res (x) n πc x n πc + x ω + x/ ω ω x It is equal to ω for x The detuning with respect to this new resonant frequency is to lowest order in x/ (x) ω ω res (x) + ω x
The radiation pressure depends on x: modifying the cavity length changes the resonance frequency, and thus (x) and F The dynamical equation for x is Mẍ MΓ M ẋ MΩ Mx + F (x) In the steady state where the derivatives vanish, we have x x s where x s (I cav ) F MΩ M SI cav MΩ (4) Mc The mirror displacement is small as compared to the wavelength At first order, Γ is also modified in principle and reads Γ Γ( x/ ) We can write (x) Γ ( + ω x + x ) Γ Γ + ω x Γ Γ + k γ x Γ + K x (5) where we used the expression of Γ and introduced K k /γ We have eventually neglected the correction to Γ, which is of order x/, smaller by a large factor ω / as compared to the correction coming from the detuning We can thus give the expression of the radiation pressure force F (x) as a function of x: F (x) AS c where F max AS /c and P (u) ( Γ Remark γc + x ( + Γ + K x ) F max P (K x) + u) is a polynomial of order two We can even write more exactly the expression taking into account the modification of Γ: the new damping rate is related to the new cavity length x + x: ) Γ (x) ( + xl On the other hand, the exact value of (x) is (x) Γ (x) Γ Γ + x/ Γ (x) Γ (x) ω ω res (x) ω We get for the exact expression of the ratio /Γ : ) ( ( + xl ω ω + x/ ω + x/ ) Γ ( ω ω + ω x ) Γ + Kx with K ω/(γ ) k/γ The exact version of Eq (5) has a k instead of k in the definition of K, we have thus just neglected a very small term K x /ω, smaller by a factor /ω (6) Using Eq (4), we can then write as a function of I cav at equilibrium: (I cav ) Γ Γ + Kx s(i cav ) Γ + KSI cav MΩ M c Γ + I cav (7) where we have introduced the characteristic intensity MΩ M c KS 3
I cav is thus the solution of an equation of degree three / f(i cav / ) where f is a polynomial of degree three: I cav A + 4 /Γ A ( + Γ + I cav ) f(icav / ) with f(u) up (u)/a A P (I cav / ) There can be several solutions for the cavity intensity I cav for a given value of This leads to a bistable behavior, with an unstable region If we write f(i cav ), the unstable region is determined by solving f (I cav ) If there is no solution or a single degenerate solution, which is the case if > 3Γ/, see below, the system is stable If there are two solutions I ± cav, the system is unstable between I + in and I in, where I± in f(i ± cav) Remark Condition of stability: the radiation pressure is a function of x, and thus derives from a potential, just like the restoring force Let us introduce the total potential V (x) corresponding to the sum of both forces Its derivative is linked to the forces F max V (x) MΩ Mx F (x) MΩ Mx P (Kx) MΩ M [ x A KP (Kx) ] [ MΩ I Mx f(kx) ] It cancels for x x s, which is solution of f(ks s ) / This equilibrium position will be stable if the second derivative of the potential is positive at this point: [ V (x) MΩ M ] + MΩ f(kx) I MKx f (Kx) f(kx) The first term cancels at the equilibrium condition, and as x s >, the condition V (x s ) > reads f (Kx s ) > Remark Study of the unstable region The steady state is obtained when f(u) / Instabilities appear if there are several solutions for u From (4), we know that the steady state is such that u > : as anticipated, the light pushes the mirror to the right We then have to look for situations where there is a flex point u > where f (u ), and such that f (u ) <, which corresponds to the situation where 3 solutions exist in a range [u, u ], the edges being defined by f (u ) f (u ) After simple arithmetics, we find that such a domain exists only if 3 < Γ (8) and u 4 /(3Γ), see Fig 3 The edges are given by ( u ) 3Γ 3Γ 4 u 3Γ ( + ) 3Γ 4 < Γ The u position is where the system becomes unstable when the intensity is increased, whereas the u position corresponds to the system being unstable when coming from above As the resonant cavity length corresponds to u res /Γ /Γ, see eg (7), the jump by decreasing intensities occurs at the left of the resonant position, for < 4
4 8 6 4 3 4 5 u u u res Figure 3: Representation of f(u), the incoming intensity in units of as a function of the mirror position in units of K, in the case Γ The points u and u, with the corresponding intensities, are the edges of the unstable region u res is the length for which the cavity is resonant If, in the case where < 3 Γ, we increase the incoming intensity starting from, we will start on the lower branch where the mirror shift is small As <, increasing the cavity length brings the cavity closer to resonance with the laser The radiation pressure thus increases more than linearly with When the reduced position reaches u, the system becomes unstable and the mirror shifts abruptly to a new position, larger than u, and which may also be larger than u res : in this case the mirror has shifted on the other side where the detuning is positive The more we increase the intensity, the more the cavity gets detuned, which ensures the system stability Now, if we lower the intensity, the mirror shifts to the left Below u, the system becomes unstable: the radiation pressure decreases too fast because the cavity gets detuned and is not able to compensate for the restoring force of the spring It jumps to a position at the left of u, where both the restoring force and the radiation pressure are small, in a quasi linear regime Quantum field in a cavity with fixed mirrors We now go to a model where the cavity field is quantized The incoming laser is close to the cavity resonance at frequency ω, and the electric field associated to a single photon in the cavity is E c hω /ε S The annihilation operator in the cavity mode is â The input laser field is in a coherent state, which complex amplitude is α in in units of E c : E in α in E c Coupling between the cavity photons and the incoming field If there is no loss in the cavity, the system can be described by a Hamiltonian: Ĥ r hω â â + hgα in e iωt â + hc The first term is the energy of the harmonic oscillator describing the quantized cavity field, â â being the total photon number operator (corresponding to I cav in the classical picture), while the second term describes a process where part of the incoming field α in e iωt is coupled through g into the cavity and creates a cavity photon The hermitian conjugate describes the reverse process If we don t make assumptions on the real character of gα in, the field hamiltonian reads ( ) Ĥ r hω â â + h gα in e iωt â + g αine iωt â 5
The cavity field state is described by a state vector ψ(t) We write the Schrödinger equation for ψ(t) : The left hand side is i h t ψ(t) e inωt c n (t) n n i h t ψ(t) Ĥr ψ(t) (9) e inωt [ i h t c n (t) + n hω c n (t)] n n We see that Ĥr n n hω n + h ( gα in e iωt n + n + + g α in eiωt n n ) The right hand side of Eq(9) is thus: Ĥ r ψ(t) + hgα in n Ĥ r ψ(t) e inωt n hω c n (t) n n e i(n+)ωt c n (t) n + n + + hg α in e i(n )ωt c n (t) n n, e inωt [ n hω c n (t) + hgα in n cn (t) + hg αin n + cn+ (t) ] n n We can therefore write for each n: i h t c n (t) n h c n (t) + hgα in n cn (t) + hg αin n + cn+ (t) It follows that the state vector in the rotating frame ψ(t) n c n(t) n obeys a Schrödinger equation for the effective Hamiltonian Cavity losses n Ĥ r h â â + hgα in â + hg α inâ () If dissipation through the coupling mirror is taken into account, the Hamiltonian description is not appropriate any more and we should switch to a description in terms of density matrices The average value of an observable  is A Tr(ˆρÂ) 3 We have by definition Then: i d a i d a i d ( {Tr(ˆρâ)} Tr i dˆρ ) â ( ) ( Tr [Ĥ h r, ˆρ] â + iγtr âˆρâ â â âˆρâ ) ˆρâ ââ Using the property of the trace, which is invariant through cyclic permutation, the second part can be recast under the form: iγtr (ˆρâ â ˆρââ â ) ˆρâ â i Γ ( [ ] ) Tr ˆρ â, â â i Γ a, 6
where we used [ â, â ] Using again the property of the trace, the first term is ) (Ĥ h Tr r ˆρâ ˆρĤ râ h ( ) [ Tr ˆρâĤ r ˆρĤ râ â, h r] Ĥ Using [â, â] and [ â, â ], the commutator reads [ ] [ ] [ â, h Ĥ r â, â â + gα in â, â ] â + gα in â + gα in After taking the average, we finally obtain i d a gα in ( + iγ/) a () 4 The steady state α cav of a is reached when d a α cav vanishes We get g + iγ/ α in It measures the total field amplitude in the cavity By analogy with the classical case, Eq (), and introducing a factor to get the total field, we can identify the coupling constant g g i t τ i γ c 5 Let use calculate the right hand side of the equation of motion of the density matrix ˆρ s We have âˆρ s â â âˆρ s ˆρ sâ â The hamiltonian term gives [ âˆρ s, â ] [ ] ˆρ s â, â α [ cav ˆρ s, â ] α cav [ˆρ s, â] ] [Ĥ h r, ˆρ s h [ ] [ ] [ ˆρ s, Ĥ r ˆρ s, â â gα in ˆρ s, â ] g αin [ˆρ s, â] We now remark that gα in α cav + iα cav Γ/ and g αin α cav iαcavγ/ The terms in Γ will thus cancel and we can write i dˆρ [ ] [ s ˆρ s, â â α cav ˆρ s, â ] αcav [ˆρ s, â] ([ ] [ ˆρ s, â â âˆρ s, â ] [ ]) ˆρ s â, â (ˆρ s â â â âˆρ s âˆρ s â + â âˆρ s ˆρ s â â + âˆρ s â ) ˆρ s is therefore a possible steady state for the density matrix 6 The density matrix associated with a pure coherent state α cav is ˆρ s α cav α cav It is clear that this density matrix verifies the two relations, as â α cav α cav α cav and α cav â α cav α cav The state towards which the cavity relaxes is a coherent state associated to the amplitude α cav 7
3 Cavity with moving mirrors: radiation cooling of the mechanical mode 3 Transition probability in the perturbative theory The coupled system {mirror + cavity field} is described by the Hamiltonian Ĥ tot Ĥ r + ˆp M + MΩ M ˆx hf ˆx ˆN, where ˆN â â and f is a constant We have added to the field Hamiltonian the quantum mechanical oscillator and the coupling arising from the radiation pressure hf ˆx ˆN corresponds to the quantum version of the classical energy E F x SI cav x/c Let us now replace I cav by its expression as a function of E c, using α cav E c E + cav (α cav describes the total field): SI cav c ε S E + cav hω E + cav E c h ω α cav h ω ˆN By identification, we can thus write f ω We describe the field by a classical fluctuating variable N(t) N +δn(t) The origin shift corresponds to chose the classical steady state x s as the new origin It corresponds to x s hfn/(mω M ) This doesn t modify the ˆp operator Writing ˆx x s + ˆx, we obtain MΩ M(x s + ˆx ) hf ˆx N hfx s N MΩ M ˆx + MΩ Mx sˆx + MΩ Mx s hf ˆx N hfx s N ( ) N MΩ M ˆx hf ˆx δn + hfx s N Apart from scalar terms, we end up for the oscillator Hamiltonian with Ĥ L hω Mˆb ˆb hf ˆx δn(t) hω Mˆb ˆb hfxm δn(t)(ˆb + ˆb ) () Remark We could instead write â α cav + δâ and â αcav + δâ, in the spirit of a Bogoliubov approach were we linearize the field around a classical field, and keep only the second order terms in the fluctuations This is what is done in Ref [], Eq() Then x s hf α cav /(MΩ M ) We also have ˆx ˆN ( (x s + ˆx ) αcav + δâ ) (α cav + δâ) α cav x s + α cav ˆx + α cav x s δâ + α cavx s δâ + α cavˆx δâ + α cavˆx δâ + x s δâ δâ where we have neglected the third order term The first term is constant, the second one cancels the shift of the harmonic oscillator center The Hamiltonian, apart from a constant term, reads Ĥ Ĥ r + hω Mˆb ˆb hf [(α cav δâ + αcavδâ)x s + x M (α cav δâ + αcavδâ)(ˆb + ˆb ] ) + x s δâ δâ The terms is x s can be reabsorbed in the definition of the detuning + fx s, as in Eq (5) This gives finally: Ĥ h δâ δâ + hω Mˆb ˆb hfxm (α cav δâ + α cavδâ)(ˆb + ˆb ) 8
3 Let us write the action of i h t, Ĥ and ĤI on ψ(t) M : i h t ψ(t) M (i h λ ) m + m hω M λ m e imω M t m, m Ĥ ψ(t) M m m hω M λ m e imω M t m, Ĥ I ψ(t) M hfδn(t) m λ m e imω M tˆx m The Schrödinger equation leads to i h λ m e imω M t m hfδn(t) λ m e im Ω tˆx M m, m m or λ m ifδn(t) λ m e i(m m)ω M t m ˆx m m We limit ourselves to the first order We should take the zeroth order for λ m δ m,m Writing the equation in the integral form, we finally have λ () m in the sum, that is λ m (t) if m ˆx m t e i(m m )Ω M t δn(t ) 4 ˆx only couples m with m ±, with m ± ˆx m x M m + ± 5 The probability to find the mirror in state m ± after a time t is 3 Fluctuation spectrum P ± (t) λ m ±(t) m + ± f x M The power spectrum density of a fluctuating signal s(t) is defined as S s (ω) lim T T where δs(t) s(t) s(t) T T δs (t)δs(t ) e iω(t t ) t e ±iω M t δn(t ) (3) dτ δs (t)δs(t + τ) e iωτ, (4) To obtain the probability at long times, we should take the statistical average and let t go to infinity We can write: t e ±iω M t δn(t ) At large t, this is equivalent by definition to t t e iω M (t t ) δn (t )δn(t ) ts N ( Ω M ) It follows that we have dp ± t m + ± f x M S N ( Ω M ) 9
The power spectrum S N of N can be shown to be S N (Ω) N Γ ( + Ω) + Γ /4 For a negative detuning, S N ( Ω M ) is smaller than S N (+Ω M ) Provided this is not compensated by the m or m + pre-factor which goes the other way round, the rate to reach a lower state P is larger than the rate for reaching a higher vibrational state As a consequence, the mirror vibration is cooled down 3 This process will stop for a average m such that the energy is stationary: Ė P + hω M P hω M We deduce the equilibrium temperature ( m + )S N ( Ω M ) m S N (Ω M ) m + ] [ hωm exp S N(Ω M ) m k B T S N ( Ω M ) ( Ω M) + Γ /4 ( + Ω M ) + Γ /4 k B T ln hω M ( ( ΩM ) +Γ /4 ( +Ω M ) +Γ /4 ) (5) The temperature is minimum if we chose Ω M + Γ 4 The average excitation number corresponding to this choice is + Γ 4Ω m M In the limit Γ Ω M where the vibrational structure is not resolved, this value corresponds to what we also find for laser cooling on a transition of wih Γ The limit temperature is k B T hγ/4 In the other limit Γ Ω M, the laser can induce Raman resolved sideband cooling, and the optimum choice is to repeat a cycle where a laser photon is absorbed and a photon of larger frequency is emitted into the cavity mode, which reduces the vibrational energy by hω M each time The limit temperature is then hω M k B T lim ( ) ln 4ΩM Γ It is much less than hω M and corresponds to a large population in the ground state: References n e hω M k B T Γ 6Ω M [] I Wilson-Rae, N Nooshi, W Zwerger, and T J Kippenberg Theory of ground state cooling of a mechanical oscillator using dynamical backaction Phys Rev Lett, 99:939, Aug 7