Supplementary Information for Thermal Noises in an Aqueous Quadrupole Micro- and Nano-Trap

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1 Supplemeary Iformaio for Thermal Noises i a Aqueous Quadrupole Micro- ad Nao-Trap Jae Hyu Park ad Predrag S. Krsić * Physics Divisio, Oak Ridge Naioal Laboraory, Oak Ridge, TN krsicp@orl.gov *correspodig auhor, Dr. Predrag S. Krsić Physics Divisio Oak Ridge Naioal Laboraory PO Box 8, Bldg. 6 Oak Ridge, TN Tel. (865) Fax (865) krsicp@orl.gov Table of Coes SI. Deailed Derivaio of Equaio (9) SI. Mahieu expoes for various b ad q SI 3. Derivaio of Equaios () SI 4. Time Hisories of xx, vv, ad xv S

2 SI. Deailed Derivaio of Equaio (9) The iegraio I(b,q, ) i equaio (8) is expressed by I b, q, s b, q, u c b, q, 4 4 bu c b, q, u s b, q, 4 4. (S) The Mahieu cosie ad sie fucios are relaed wih he Floque soluio of Mahieu equaio, F as c,,,, F F a q F a q, (Sa) s,,,, F F a q F a q. (Sb) Hereafer, for breviy equaio, F, has he form of a b 4 uless specified oherwise. Sice he Floque soluio of Mahieu,, exp i P F a q. (S3) where μ is Mahieu expoe (complex umber) ad P is he complex valued fucio which is periodic wih period π. From equaios (S) ad (S), c ad s ca be expressed as: exp i P exp i P c F s i Pa q i Pa q F exp,, exp,,, (S4a). (S4b) By iserig equaios (S3) io equaio (S), he we obai S

3 4 F F I exp,,,,,,,, exp b i u P u P b u P a q u P a q u P a q P a q exp b i u P u P,,,,,, I a q I a q I a q 3. (S5) Rewriig =+ ( <, is o-zero ieger) ad usig he divisio propery of iegraio,, (S6) ad he π periodiciy of P, i.e. P P, I, I, I i equaio (B4) ca be compued as I l 3 bi l bi u,,,,,,,, e e P a q u P a q biu e P a q u P a q bi e biu e P u P b i e biu e P u P, (S7a) I l,,,,,,,, b l b u e e P a q u P a q u P a q P a q,,,,,,,, bu e P a q u P a q u P a q P a q b e bu e P u P u P P b e,,,,,,,, bu e P a q u P a q u P a q P a q dv, (S7b) S3

4 I 3 l bi l bi u,,,,,,,, e e P a q u P a q biu e P a q u P a q bi e biu e P u P b i e biv e P v P dv. (S7c) Thereby,,, I a q ca be wrie as: I I Ia Ib Ic 4, (S8a),,,,,,,,,, bu, (S8b) I a q s a q u c a q c a q u s a q bi e bu bi Ia e, (S8c) F u F F F F u F u F F b e bu Ib b, q, b e F F, (S8d) bi e bu F a q bi Ic b, q, e. (S8e),, u F F F The equaios (S8b-e) prese he iegraios I b, q,, I a b, q,, b,, I c b, q, i equaio (9). I order o achieve he sable log-ime behavior of I I b q, ad, all he expressios of equaios (S8c)-(S8e) eed o coverge ad he expoeial erm e -(b-iμ) should vaish for. The ecessary codiio for his is b where is he imagiary par of Mahieu expoe, μ=+i. I is same as he codiio ha he mea moio of paricle is sable [S]. The log-ime behaviors of I a, I b, ad I c i sable codiio ca be obaied wih, S4

5 bu Ia bi e, (S9a) F u F F F F u F u F F bu Ib b e F F,(S9b) bu Ic bi e, (S9c) F u F F F SI. Mahieu Expoes for Various b ad q I he aid of figure S, we ca compue he q-rage saisfyig he codiio of <<b/ for q give b. Figure S. Variaio of Mahieu expoe wih q for a give b. ad are he real ad imagiary pars of Mahieu expoe, respecively. S5

6 SI 3. Derivaio of Equaios () The iegraio J(b,q, ) i equaio () is origially give by b J b, q, s b, q, u c b, q, s b, q, u c b, q, (S) b bu c b, q, u s b, q, c b, q, u s b, q, The iegraio of J(b,q, ), equaio (S), ca be re-wrie as he summaio of hree erms: b J J bj J3. (Sa) 4 wih,,,,,,,,,, bu. (Sb) J a q s a q u c a q c a q u s a q,,,,,,,,,, J a q s a q u c a q c a q u s a q. (Sc) b u s u c c u s,,,,,,,,,, bu. (Sd) J a q s a q u c a q c a q u s a q 3 I exac Here a b 4. Ieresigly, J 3 (b,q, ) is equal o I(b,q, ) i he flucuaio of posiio. Followig he same procere for he flucuaio of posiios i S, we ca wrie he ime as = + ( <, is o-zero ieger) ad he, he each erm ca be wrie as follow: For J (b,q, ), J a q J a q Ja a q Jb a q Jc a q 4,,,,,,,,,,, (Sa) wih,,,,,,,,,, bu, (Sb) J a q s a q u c a q c a q u s a q bi e bv F u F Ja bi e, (Sc) F F S6

7 b e bv F u F F F u b Jb e F F F F, (Sd) bi e bv F u F Jc bi e. (Se) For J (b,q, ), J a q J a q Ja a q Jb a q Jc a q 4,,,,,,,,,,, (S3a) wih,,,,,,,,,,, (S3b) J a q s a q u c a q c a q u s a q bu,,,,,,,, s a q u c a q c a q u s a q F u F F bv e, (S3c) F F bi e a,, bi J a q J b b e b e e F u F u F F F F F F bu, (S3d) bi e bv c,, bi F u F F J a q e F F. (S3e) For J 3 (b,q, ), J3 J3 J3a J3b J3c 4, (S4a) wih 3,,,,,,,,,, bu, (S4b) J a q s a q u c a q c a q u s a q S7

8 bi e bu bi J3a, (S4c) e F u F F F F u F u F F b e bu J3b b e F F, (S4d) bi e bu F a q bi J3c. (S4e) e,, u F F F Collecig all he expressios i above, equaio (Sa) ca be wrie i he followig simple forms as below: b J J bj J3, (Sa) 4 For J (b,q, ), J b q J b q J b q J b q J b q 4,,,, a,, b,, c,,, (S5a) wih bi a,, a,, J b q J b q e, (S5b) b,, b,, J b q J b q e, (S5c) b bi c,, c,, J b q J b q e, (S5d),,,,,,,,,, bu, (S5e) J a q s a q u c a q c a q u s a q bu Ja bi, (S5f) e F u F F F F u F u F F bu Ja b e F F, (S5g) S8

9 ad bu Ja bi. (S5h) e F u F F F For J (b,q, ), J b q J b q J b q J b q J b q 4,,,, a,, b,, c,,, (S6a) wih bi a,, a,, J b q J b q e, (S6b) b,, b,, J b q J b q e, (S6c) b bi c,, c,, J b q J b q e, (S6d),,,,,,,,,,, (S6e) bu J a q s a q u c a q c a q u s a q,,,,,,,, s a q u c a q c a q u s a q F v F F bv Ja e dv bi e F F, (S6f) F v F v F F F F bv Jb e dv b e F F, (S6g) ad F v F F bv Jc e dv bi e F F. (S6h) For J 3 (b,q, ), J3 b q J3 b q J3 b q J3 b q J3 b q 4,,,, a,, b,, c,,, (S7a) wih S9

10 3 bi 3a,, 3a,, J b q J a q e, (S7b) b,, b,, J b q J a q e, (S7c) 3b 3 bi 3c,, 3c,, J b q J a q e, (S7d),,,,,,,,,, bu, (S7e) J a q s a q u c a q c a q u s a q bu J3a bi, (S7f) e F u F F F F u F u F F bu J3b b e F F, (S7g) ad bu J3c bi. (S7h) e F u F F F S

11 SI 4. Time Hisories of xx, vv, ad xv for large b = 4. Figure S. Time hisories of xx, vv, ad xv for b=.: (a), (b), (c) for q = 3.; (d), (e), (f) for q =4.. Figures S show he rasie behaviour of xx, vv, ad xv for b=4. ad q = 3. (a, b, c) ad for b=4. ad q = 4. (d, e, f). If q furher icreases (q > 4), he posiio flucuaio xx is divergig alog wih vv, ad xv ad he fourh frequecy mode sars o coribue i addiio o he firs hree modes, which domiaes for q=3.. I should be oed ha he velociy flucuaios are icreased more ha posiio ad covariace flucuaios, which idicaes ha he divergece of he posiio flucuaio origiaes from he velociy ad propagaes o he posiio divergece wih he aid of covariace. S

David Randall. ( )e ikx. k = u x,t. u( x,t)e ikx dx L. x L /2. Recall that the proof of (1) and (2) involves use of the orthogonality condition.

David Randall. ( )e ikx. k = u x,t. u( x,t)e ikx dx L. x L /2. Recall that the proof of (1) and (2) involves use of the orthogonality condition. ! Revised April 21, 2010 1:27 P! 1 Fourier Series David Radall Assume ha u( x,) is real ad iegrable If he domai is periodic, wih period L, we ca express u( x,) exacly by a Fourier series expasio: ( ) =